Coordinate geometry

Tutormate > CBSE Syllabus-Class 10th Maths > Coordinate geometry

07 COORDINATE GEOMETRY

X- coordinate(Abscissa) : It is the distance of a point from Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeywaaaa@37D0@ axis.

 

Y- coordinate(Ordinate) : It is the distance of a point from X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeiwaaaa@37CF@ axis.

 

Distance formula: Given points A( x 1 , y 1 ),   B( x 2 , y 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyqamaabm aabaGaaeiEamaaBaaaleaacaqGXaaabeaakiaabYcacaqGGaGaaeyE amaaBaaaleaacaqGXaaabeaaaOGaayjkaiaawMcaaiaabYcacaqGGa GaaeiiaiaabccacaqGcbWaaeWaaeaacaqG4bWaaSbaaSqaaiaabkda aeqaaOGaaeilaiaabccacaqG5bWaaSbaaSqaaiaabkdaaeqaaaGcca GLOaGaayzkaaaaaa@4765@

The distance from A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeyqaaaa@37B8@ to B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOqaaaa@37B9@ is d( A, B ) =  ( x 2 - x 1 ) 2 ( y 2 - y 1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeizamaabm aabaGaaeyqaiaabYcacaqGGaGaaeOqaaGaayjkaiaawMcaaiaabcca caqG9aGaaeiiamaakaaabaWaaeWaaeaacaqG4bWaaSbaaSqaaiaabk daaeqaaOGaaeylaiaabccacaqG4bWaaSbaaSqaaiaabgdaaeqaaaGc caGLOaGaayzkaaWaaWbaaSqabeaacaqGYaaaaOGaae4kaiaabccada qadaqaaiaabMhadaWgaaWcbaGaaeOmaaqabaGccaqGTaGaaeiiaiaa bMhadaWgaaWcbaGaaeymaaqabaaakiaawIcacaGLPaaadaahaaWcbe qaaiaabkdaaaaabeaaaaa@4DC6@

 

Section Formula:Given points A( x 1 , y 1 ), B( x 2 , y 2 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyqamaabm aabaGaaeiEamaaBaaaleaacaqGXaaabeaakiaabYcacaqGGaGaaeyE amaaBaaaleaacaqGXaaabeaaaOGaayjkaiaawMcaaiaabYcacaqGGa GaaeOqamaabmaabaGaaeiEamaaBaaaleaacaqGYaaabeaakiaabYca caqGGaGaaeyEamaaBaaaleaacaqGYaaabeaaaOGaayjkaiaawMcaai aac6caaaa@46D1@ if point P( x, y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeiua8aadaqadaqaa8qacaqG4bGaaeilaiaabccacaqG5baapaGa ayjkaiaawMcaaaaa@3CC7@ which divides the line segment AB MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeyqaiaabkeaaaa@387D@ internally in the ratio m:n, then coordinates of points P MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeiuaaaa@37C7@ are given by

P = ( mx 2 + nx 1 m + n my 2 + ny 1 m + n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiuaiaabc cacaqG9aGaaeiiamaabmaabaWaaSaaaeaacaqGTbGaaeiEamaaBaaa leaacaqGYaaabeaakiaabUcacaqGGaGaaeOBaiaabIhadaWgaaWcba GaaeymaaqabaaakeaacaqGTbGaaeiiaiaabUcacaqGGaGaaeOBaaaa caqGSaGaaeiiamaalaaabaGaaeyBaiaabMhadaWgaaWcbaGaaeOmaa qabaGccaqGRaGaaeiiaiaab6gacaqG5bWaaSbaaSqaaiaabgdaaeqa aaGcbaGaaeyBaiaabccacaqGRaGaaeiiaiaab6gaaaaacaGLOaGaay zkaaaaaa@5170@

 

Mid point formula: If P is mid point of segment AB,where A( x 1 , y 1 ), B( x 2 , y 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyqamaabm aabaGaaeiEamaaBaaaleaacaqGXaaabeaakiaabYcacaqGGaGaaeyE amaaBaaaleaacaqGXaaabeaaaOGaayjkaiaawMcaaiaabYcacaqGGa GaaeOqamaabmaabaGaaeiEamaaBaaaleaacaqGYaaabeaakiaabYca caqGGaGaaeyEamaaBaaaleaacaqGYaaabeaaaOGaayjkaiaawMcaaa aa@461F@

Then P = ( x 2 + x 1 2 y 2 + y 1 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiuaiaabc cacaqG9aGaaeiiamaabmaabaWaaSaaaeaacaqG4bWaaSbaaSqaaiaa bkdaaeqaaOGaae4kaiaabccacaqG4bWaaSbaaSqaaiaabgdaaeqaaa GcbaGaaeOmaaaacaqGSaGaaeiiamaalaaabaGaaeyEamaaBaaaleaa caqGYaaabeaakiaabUcacaqGGaGaaeyEamaaBaaaleaacaqGXaaabe aaaOqaaiaabkdaaaaacaGLOaGaayzkaaaaaa@476E@

 

Area of triangle : If A( x 1 , y 1 ), B( x 2 , y 2 ), C( x 3 , y 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyqamaabm aabaGaaeiEamaaBaaaleaacaqGXaaabeaakiaabYcacaqGGaGaaeyE amaaBaaaleaacaqGXaaabeaaaOGaayjkaiaawMcaaiaabYcacaqGGa GaaeOqamaabmaabaGaaeiEamaaBaaaleaacaqGYaaabeaakiaabYca caqGGaGaaeyEamaaBaaaleaacaqGYaaabeaaaOGaayjkaiaawMcaai aabYcacaqGGaGaae4qamaabmaabaGaaeiEamaaBaaaleaacaqGZaaa beaakiaabYcacaqGGaGaaeyEamaaBaaaleaacaqGZaaabeaaaOGaay jkaiaawMcaaaaa@4EE1@ are vertices of triangle then area of triangle is given by

1 2 x 1 ( y 2 - y 3 )  + x 2 ( y 3 - y 1 )  + x 3 ( y 1 - y 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGOmaaaadaGcaaqaaiaabIhadaWgaaWcbaGaaeymaaqa baGcdaqadaqaaiaabMhadaWgaaWcbaGaaeOmaaqabaGccaqGTaGaae iiaiaabMhadaWgaaWcbaGaae4maaqabaaakiaawIcacaGLPaaacaqG GaGaae4kaiaabccacaqG4bWaaSbaaSqaaiaabkdaaeqaaOWaaeWaae aacaqG5bWaaSbaaSqaaiaabodaaeqaaOGaaeylaiaabccacaqG5bWa aSbaaSqaaiaabgdaaeqaaaGccaGLOaGaayzkaaGaaeiiaiaabUcaca qGGaGaaeiEamaaBaaaleaacaqGZaaabeaakmaabmaabaGaaeyEamaa BaaaleaacaqGXaaabeaakiaab2cacaqGGaGaaeyEamaaBaaaleaaca qGYaaabeaaaOGaayjkaiaawMcaaaWcbeaaaaa@5530@

 

* To find the area of a polygon we divide it in triangles and take numerical value of the area of each of the triangles.

 

* The area of ΔABC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaae yqaiaabkeacaqGdbaaaa@39AC@ can also be computed by using the following steps:

STEP I: Write the coordinates of the vertices A( x 1 , y 1 ), B( x 2 , y 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyqamaabm aabaGaaeiEamaaBaaaleaacaqGXaaabeaakiaabYcacaqGGaGaaeyE amaaBaaaleaacaqGXaaabeaaaOGaayjkaiaawMcaaiaabYcacaqGGa GaaeOqamaabmaabaGaaeiEamaaBaaaleaacaqGYaaabeaakiaabYca caqGGaGaaeyEamaaBaaaleaacaqGYaaabeaaaOGaayjkaiaawMcaaa aa@461F@ and C( x 3 , y 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4qamaabm aabaGaaeiEamaaBaaaleaacaqGZaaabeaakiaabYcacaqGGaGaaeyE amaaBaaaleaacaqGZaaabeaaaOGaayjkaiaawMcaaaaa@3D66@ in three

columns as shown below and augment the coordinates of A( x 1 , y 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyqamaabm aabaGaaeiEamaaBaaaleaacaqGXaaabeaakiaabYcacaqGGaGaaeyE amaaBaaaleaacaqGXaaabeaaaOGaayjkaiaawMcaaaaa@3D60@ as fourth column.

 

STEP II:Draw broken parallel lines pointing down wards from left to right and right to left.

x 1 x 2 x 3 x 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiEamaaBa aaleaacaqGXaaabeaakiaabIhadaWgaaWcbaGaaeOmaaqabaGccaqG 4bWaaSbaaSqaaiaabodaaeqaaOGaaeiEamaaBaaaleaacaqGXaaabe aaaaa@3D83@

 

y 1 y 2 y 3 y 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyEamaaBa aaleaacaqGXaaabeaakiaabMhadaWgaaWcbaGaaeOmaaqabaGccaqG 5bWaaSbaaSqaaiaabodaaeqaaOGaaeyEamaaBaaaleaacaqGXaaabe aaaaa@3D87@

 

STEP III:Compute the sum of the products of numbers at the ends of lines pointing

downwards from left to right and subtract from this sum, the sum of the products of numbers at the ends of the lines pointing downward from right to left i.e., compute

( x 1 y 2 + x 2 y 3 + x 3 y 1 ) - ( x 2 y 1 + x 3 y 2 + x 1 y 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca qG4bWaaSbaaSqaaiaabgdaaeqaaOGaaeyEamaaBaaaleaacaqGYaaa beaakiaabUcacaqGGaGaaeiEamaaBaaaleaacaqGYaaabeaakiaabM hadaWgaaWcbaGaae4maaqabaGccaqGRaGaaeiiaiaabIhadaWgaaWc baGaae4maaqabaGccaqG5bWaaSbaaSqaaiaabgdaaeqaaaGccaGLOa GaayzkaaGaaeiiaiaab2cacaqGGaWaaeWaaeaacaqG4bWaaSbaaSqa aiaabkdaaeqaaOGaaeyEamaaBaaaleaacaqGXaaabeaakiaabUcaca qGGaGaaeiEamaaBaaaleaacaqGZaaabeaakiaabMhadaWgaaWcbaGa aeOmaaqabaGccaqGRaGaaeiiaiaabIhadaWgaaWcbaGaaeymaaqaba GccaqG5bWaaSbaaSqaaiaabodaaeqaaaGccaGLOaGaayzkaaaaaa@5710@

 

STEP IV: Find the absolute of the number obtained in step III and take its half to obtain the area.

 

* Three points A( x 1 , y 1 ), B( x 2 , y 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyqamaabm aabaGaaeiEamaaBaaaleaacaqGXaaabeaakiaabYcacaqGGaGaaeyE amaaBaaaleaacaqGXaaabeaaaOGaayjkaiaawMcaaiaabYcacaqGGa GaaeOqamaabmaabaGaaeiEamaaBaaaleaacaqGYaaabeaakiaabYca caqGGaGaaeyEamaaBaaaleaacaqGYaaabeaaaOGaayjkaiaawMcaaa aa@461F@ and C( x 3 , y 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4qamaabm aabaGaaeiEamaaBaaaleaacaqGZaaabeaakiaabYcacaqGGaGaaeyE amaaBaaaleaacaqGZaaabeaaaOGaayjkaiaawMcaaaaa@3D66@ are collinear if

Area of ΔABC = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiLdiaabg eacaqGcbGaae4qaiaabccacaqG9aGaaeiiaiaabcdaaaa@3C18@ i.e., x 1 ( y 2 - y 3 )  + x 2 ( y 3 - y 1 )  + x 3 ( y 1 - y 2 ) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiEamaaBa aaleaacaqGXaaabeaakmaabmaabaGaaeyEamaaBaaaleaacaqGYaaa beaakiaab2cacaqGGaGaaeyEamaaBaaaleaacaqGZaaabeaaaOGaay jkaiaawMcaaiaabccacaqGRaGaaeiiaiaabIhadaWgaaWcbaGaaeOm aaqabaGcdaqadaqaaiaabMhadaWgaaWcbaGaae4maaqabaGccaqGTa GaaeiiaiaabMhadaWgaaWcbaGaaeymaaqabaaakiaawIcacaGLPaaa caqGGaGaae4kaiaabccacaqG4bWaaSbaaSqaaiaabodaaeqaaOWaae WaaeaacaqG5bWaaSbaaSqaaiaabgdaaeqaaOGaaeylaiaabccacaqG 5bWaaSbaaSqaaiaabkdaaeqaaaGccaGLOaGaayzkaaGaaeiiaiaab2 dacaqGGaGaaeimaaaa@5647@

 

 

 

THEOREM 1: Prove that the coordinates of the centroid of the triangle whose vertices are

( x 1 , y 1 ),( x 2 , y 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca qG4bWaaSbaaSqaaiaabgdaaeqaaOGaaeilaiaabccacaqG5bWaaSba aSqaaiaabgdaaeqaaaGccaGLOaGaayzkaaGaaeilamaabmaabaGaae iEamaaBaaaleaacaqGYaaabeaakiaabYcacaqGGaGaaeyEamaaBaaa leaacaqGYaaabeaaaOGaayjkaiaawMcaaaaa@43F3@ and ( x 3 , y 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca qG4bWaaSbaaSqaaiaabodaaeqaaOGaaeilaiaabccacaqG5bWaaSba aSqaaiaabodaaeqaaaGccaGLOaGaayzkaaaaaa@3CA0@ are

 

( x 1  + x 2  + x 3 3 y 1  + y 2  + y 3 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaabIhadaWgaaWcbaGaaeymaaqabaGccaqGGaGaae4kaiaa bccacaqG4bWaaSbaaSqaaiaabkdaaeqaaOGaaeiiaiaabUcacaqGGa GaaeiEamaaBaaaleaacaqGZaaabeaaaOqaaiaabodaaaGaaeilaiaa bccadaWcaaqaaiaabMhadaWgaaWcbaGaaeymaaqabaGccaqGGaGaae 4kaiaabccacaqG5bWaaSbaaSqaaiaabkdaaeqaaOGaaeiiaiaabUca caqGGaGaaeyEamaaBaaaleaacaqGZaaabeaaaOqaaiaabodaaaaaca GLOaGaayzkaaaaaa@4D94@

Also deduce that the medians of a triangle are concurrent.

[NCERT, CBSE 2004]

PROOF: Let A( x 1 , y 1 ),   B( x 2 , y 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyqamaabm aabaGaaeiEamaaBaaaleaacaqGXaaabeaakiaabYcacaqGGaGaaeyE amaaBaaaleaacaqGXaaabeaaaOGaayjkaiaawMcaaiaabYcacaqGGa GaaeiiaiaabccacaqGcbWaaeWaaeaacaqG4bWaaSbaaSqaaiaabkda aeqaaOGaaeilaiaabccacaqG5bWaaSbaaSqaaiaabkdaaeqaaaGcca GLOaGaayzkaaaaaa@4765@ and C( x 3 , y 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4qamaabm aabaGaaeiEamaaBaaaleaacaqGZaaabeaakiaabYcacaqGGaGaaeyE amaaBaaaleaacaqGZaaabeaaaOGaayjkaiaawMcaaaaa@3D66@ be the vertices of ΔABC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaae yqaiaabkeacaqGdbaaaa@39AC@ whose medians are AD, BE and CF MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4qaiaabA eaaaa@3785@ respectively. So D, E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiraiaabY cacaqGGaGaaeyraaaa@38D7@ and F MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOraaaa@36BF@ are respectively the mid-points of BC, CA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOqaiaabo eacaqGSaGaaeiiaiaaboeacaqGbbaaaa@3A5D@ and AB MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyqaiaabk eaaaa@377F@ .

Coordinates of D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGebaaaa@36DD@ are ( x 2  + x 3 2 y 2  + y 3 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaabIhadaWgaaWcbaGaaeOmaaqabaGccaqGGaGaae4kaiaa bccacaqG4bWaaSbaaSqaaiaabodaaeqaaaGcbaGaaeOmaaaacaqGSa GaaeiiamaalaaabaGaaeyEamaaBaaaleaacaqGYaaabeaakiaabcca caqGRaGaaeiiaiaabMhadaWgaaWcbaGaae4maaqabaaakeaacaqGYa aaaaGaayjkaiaawMcaaaaa@45DF@

Coordinates of a point dividing AD MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGbbGaaeiraaaa@37A1@ in the ratio 2:1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIYaGaaiOoaiaaigdaaaa@384B@ are

( 1x 1 + 2( x 2  + x 3 2 ) 1 + 2 ,    1y 1 + 2( y 2  + y 3 2 ) 1 + 2 ) = ( x 1  + x 2  + x 3 3 y 1  + y 2  + y 3 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaabgdacaqG4bWaaSbaaSqaaiaabgdaaeqaaOGaae4kaiaa bccacaqGYaWaaeWaaeaadaWcaaqaaiaabIhadaWgaaWcbaGaaeOmaa qabaGccaqGGaGaae4kaiaabccacaqG4bWaaSbaaSqaaiaabodaaeqa aaGcbaGaaeOmaaaaaiaawIcacaGLPaaaaeaacaqGXaGaaeiiaiaabU cacaqGGaGaaeOmaaaacaqGSaGaaeiiaiaabccacaqGGaWaaSaaaeaa caqGXaGaaeyEamaaBaaaleaacaqGXaaabeaakiaabUcacaqGGaGaae OmamaabmaabaWaaSaaaeaacaqG5bWaaSbaaSqaaiaabkdaaeqaaOGa aeiiaiaabUcacaqGGaGaaeyEamaaBaaaleaacaqGZaaabeaaaOqaai aabkdaaaaacaGLOaGaayzkaaaabaGaaeymaiaabccacaqGRaGaaeii aiaabkdaaaaacaGLOaGaayzkaaGaaeiiaiaab2dacaqGGaWaaeWaae aadaWcaaqaaiaabIhadaWgaaWcbaGaaeymaaqabaGccaqGGaGaae4k aiaabccacaqG4bWaaSbaaSqaaiaabkdaaeqaaOGaaeiiaiaabUcaca qGGaGaaeiEamaaBaaaleaacaqGZaaabeaaaOqaaiaabodaaaGaaeil aiaabccadaWcaaqaaiaabMhadaWgaaWcbaGaaeymaaqabaGccaqGGa Gaae4kaiaabccacaqG5bWaaSbaaSqaaiaabkdaaeqaaOGaaeiiaiaa bUcacaqGGaGaaeyEamaaBaaaleaacaqGZaaabeaaaOqaaiaabodaaa aacaGLOaGaayzkaaaaaa@73F4@

The coordinates of E are ( x 1  + x 3 2 y 1  + y 3 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaabIhadaWgaaWcbaGaaeymaaqabaGccaqGGaGaae4kaiaa bccacaqG4bWaaSbaaSqaaiaabodaaeqaaaGcbaGaaeOmaaaacaqGSa GaaeiiamaalaaabaGaaeyEamaaBaaaleaacaqGXaaabeaakiaabcca caqGRaGaaeiiaiaabMhadaWgaaWcbaGaae4maaqabaaakeaacaqGYa aaaaGaayjkaiaawMcaaaaa@45DD@

The coordinates of a point dividing BE in the ratio 2:1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIYaGaaiOoaiaaigdaaaa@384B@ are

( 1x 2 + 2( x 1  + x 3 2 ) 1 + 2 1y 2 + 2( y 1  + y 3 2 ) 1 + 2 ) = ( x 1  + x 2  + x 3 3 y 1  + y 2  + y 3 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaabgdacaqG4bWaaSbaaSqaaiaabkdaaeqaaOGaae4kaiaa bccacaqGYaWaaeWaaeaadaWcaaqaaiaabIhadaWgaaWcbaGaaeymaa qabaGccaqGGaGaae4kaiaabccacaqG4bWaaSbaaSqaaiaabodaaeqa aaGcbaGaaeOmaaaaaiaawIcacaGLPaaaaeaacaqGXaGaaeiiaiaabU cacaqGGaGaaeOmaaaacaqGSaGaaeiiamaalaaabaGaaeymaiaabMha daWgaaWcbaGaaeOmaaqabaGccaqGRaGaaeiiaiaabkdadaqadaqaam aalaaabaGaaeyEamaaBaaaleaacaqGXaaabeaakiaabccacaqGRaGa aeiiaiaabMhadaWgaaWcbaGaae4maaqabaaakeaacaqGYaaaaaGaay jkaiaawMcaaaqaaiaabgdacaqGGaGaae4kaiaabccacaqGYaaaaaGa ayjkaiaawMcaaiaabccacaqG9aGaaeiiamaabmaabaWaaSaaaeaaca qG4bWaaSbaaSqaaiaabgdaaeqaaOGaaeiiaiaabUcacaqGGaGaaeiE amaaBaaaleaacaqGYaaabeaakiaabccacaqGRaGaaeiiaiaabIhada WgaaWcbaGaae4maaqabaaakeaacaqGZaaaaiaabYcacaqGGaWaaSaa aeaacaqG5bWaaSbaaSqaaiaabgdaaeqaaOGaaeiiaiaabUcacaqGGa GaaeyEamaaBaaaleaacaqGYaaabeaakiaabccacaqGRaGaaeiiaiaa bMhadaWgaaWcbaGaae4maaqabaaakeaacaqGZaaaaaGaayjkaiaawM caaaaa@72AE@

Similarly the coordinates of a point dividing CF in the ratio 2:1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIYaGaaiOoaiaaigdaaaa@384B@ are

( x 1  + x 2  + x 3 3 y 1  + y 2  + y 3 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaabIhadaWgaaWcbaGaaeymaaqabaGccaqGGaGaae4kaiaa bccacaqG4bWaaSbaaSqaaiaabkdaaeqaaOGaaeiiaiaabUcacaqGGa GaaeiEamaaBaaaleaacaqGZaaabeaaaOqaaiaabodaaaGaaeilaiaa bccadaWcaaqaaiaabMhadaWgaaWcbaGaaeymaaqabaGccaqGGaGaae 4kaiaabccacaqG5bWaaSbaaSqaaiaabkdaaeqaaOGaaeiiaiaabUca caqGGaGaaeyEamaaBaaaleaacaqGZaaabeaaaOqaaiaabodaaaaaca GLOaGaayzkaaaaaa@4D94@ .

Thus, the point having coordinates ( x 1  + x 2  + x 3 3 y 1  + y 2  + y 3 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaabIhadaWgaaWcbaGaaeymaaqabaGccaqGGaGaae4kaiaa bccacaqG4bWaaSbaaSqaaiaabkdaaeqaaOGaaeiiaiaabUcacaqGGa GaaeiEamaaBaaaleaacaqGZaaabeaaaOqaaiaabodaaaGaaeilaiaa bccadaWcaaqaaiaabMhadaWgaaWcbaGaaeymaaqabaGccaqGGaGaae 4kaiaabccacaqG5bWaaSbaaSqaaiaabkdaaeqaaOGaaeiiaiaabUca caqGGaGaaeyEamaaBaaaleaacaqGZaaabeaaaOqaaiaabodaaaaaca GLOaGaayzkaaaaaa@4D94@ is common to AD, BE MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGbbGaaeiraiaabYcacaqGGaGaaeOqaiaabweaaaa@3A80@ and CF MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGdbGaaeOraaaa@37A5@ and divides them in the ratio 1:2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIXaGaaiOoaiaaikdaaaa@384B@ .

Hence, medians of a triangle are concurrent and the coordinates of the centroid are ( x 1  + x 2  + x 3 3 y 1  + y 2  + y 3 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaabIhadaWgaaWcbaGaaeymaaqabaGccaqGGaGaae4kaiaa bccacaqG4bWaaSbaaSqaaiaabkdaaeqaaOGaaeiiaiaabUcacaqGGa GaaeiEamaaBaaaleaacaqGZaaabeaaaOqaaiaabodaaaGaaeilaiaa bccadaWcaaqaaiaabMhadaWgaaWcbaGaaeymaaqabaGccaqGGaGaae 4kaiaabccacaqG5bWaaSbaaSqaaiaabkdaaeqaaOGaaeiiaiaabUca caqGGaGaaeyEamaaBaaaleaacaqGZaaabeaaaOqaaiaabodaaaaaca GLOaGaayzkaaaaaa@4D94@

 

 

 

 

THEOREM 2:The area of a triangle, the coordinates of whose vertices are ( x 1 , y 1 ),  ( x 2 , y 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca qG4bWaaSbaaSqaaiaabgdaaeqaaOGaaeilaiaabccacaqG5bWaaSba aSqaaiaabgdaaeqaaaGccaGLOaGaayzkaaGaaeilaiaabccacaqGGa WaaeWaaeaacaqG4bWaaSbaaSqaaiaabkdaaeqaaOGaaeilaiaabcca caqG5bWaaSbaaSqaaiaabkdaaeqaaaGccaGLOaGaayzkaaaaaa@4539@ and ( x 3 , y 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca qG4bWaaSbaaSqaaiaabodaaeqaaOGaaeilaiaabccacaqG5bWaaSba aSqaaiaabodaaeqaaaGccaGLOaGaayzkaaaaaa@3CA0@ is 1 2 | x 1 ( y 2  - y 3 )  + x 2 ( y 3  - y 1 )  + x 3 ( y 1  - y 2 ) | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca qGXaaabaGaaeOmaaaadaabdaqaaiaabIhadaWgaaWcbaGaaeymaaqa baGcdaqadaqaaiaabMhadaWgaaWcbaGaaeOmaaqabaGccaqGGaGaae ylaiaabccacaqG5bWaaSbaaSqaaiaabodaaeqaaaGccaGLOaGaayzk aaGaaeiiaiaabUcacaqGGaGaaeiEamaaBaaaleaacaqGYaaabeaakm aabmaabaGaaeyEamaaBaaaleaacaqGZaaabeaakiaabccacaqGTaGa aeiiaiaabMhadaWgaaWcbaGaaeymaaqabaaakiaawIcacaGLPaaaca qGGaGaae4kaiaabccacaqG4bWaaSbaaSqaaiaabodaaeqaaOWaaeWa aeaacaqG5bWaaSbaaSqaaiaabgdaaeqaaOGaaeiiaiaab2cacaqGGa GaaeyEamaaBaaaleaacaqGYaaabeaaaOGaayjkaiaawMcaaaGaay5b SlaawIa7aaaa@5A12@

 

PROOF: LET ABC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGbbGaaeOqaiaaboeaaaa@3865@ be a triangle whose vertices are A( x 1 , y 1 ), B( x 2 , y 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyqamaabm aabaGaaeiEamaaBaaaleaacaqGXaaabeaakiaabYcacaqGGaGaaeyE amaaBaaaleaacaqGXaaabeaaaOGaayjkaiaawMcaaiaabYcacaqGGa GaaeOqamaabmaabaGaaeiEamaaBaaaleaacaqGYaaabeaakiaabYca caqGGaGaaeyEamaaBaaaleaacaqGYaaabeaaaOGaayjkaiaawMcaaa aa@461F@ and C( x 3 , y 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4qamaabm aabaGaaeiEamaaBaaaleaacaqGZaaabeaakiaabYcacaqGGaGaaeyE amaaBaaaleaacaqGZaaabeaaaOGaayjkaiaawMcaaaaa@3D66@ . Draw AL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGbbGaaeitaaaa@37A9@ ,

BM and CN MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGdbGaaeOtaaaa@37AD@ perpendiculars from A, B, C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGbbGaaeilaiaabccacaqGcbGaaeilaiaabccacaqGdbaaaa@3B09@ on the x - MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqG4bGaaeiiaiaab2caaaa@3864@ axis.

 

Clearly, ABML, ALNC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGbbGaaeOqaiaab2eacaqGmbGaaeilaiaabccacaqGbbGaaeit aiaab6eacaqGdbaaaa@3DBA@ and BMNC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGcbGaaeytaiaab6eacaqGdbaaaa@3942@ are all trapeziums.

 

We know that

Area of trapezium 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeypaiaabc cadaWcaaqaaiaabgdaaeaacaqGYaaaaaaa@38D2@ (Sum of parallel sides)(Distance between them)

We have,

Area of ΔΑΒC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaae yKdiaabk5acaqGdbaaaa@3A52@ = Area of trapezium ABML + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyqaiaabk eacaqGnbGaaeitaiaabccacaqGRaaaaa@3A6F@ Area of trapezium ALNC - MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyqaiaabY eacaqGobGaae4qaiaabccacaqGTaaaaa@3A73@ Area of trapezium BMNC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGcbGaaeytaiaab6eacaqGdbaaaa@3942@ Let Δ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqeaaa@375D@ denote the area of ΔΑΒC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaae yKdiaabk5acaqGdbaaaa@3A52@ . Then,

Δ =  1 2 ( BM + AL )( ML ) +  1 2 ( AL + CN )( LN ) -  1 2 ( BM + CN )( MN ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiLdiaabc cacaqG9aGaaeiiamaalaaabaGaaeymaaqaaiaabkdaaaWaaeWaaeaa caqGcbGaaeytaiaabccacaqGRaGaaeiiaiaabgeacaqGmbaacaGLOa GaayzkaaWaaeWaaeaacaqGnbGaaeitaaGaayjkaiaawMcaaiaabcca caqGRaGaaeiiamaalaaabaGaaeymaaqaaiaabkdaaaWaaeWaaeaaca qGbbGaaeitaiaabccacaqGRaGaaeiiaiaaboeacaqGobaacaGLOaGa ayzkaaWaaeWaaeaacaqGmbGaaeOtaaGaayjkaiaawMcaaiaabccaca qGTaGaaeiiamaalaaabaGaaeymaaqaaiaabkdaaaWaaeWaaeaacaqG cbGaaeytaiaabccacaqGRaGaaeiiaiaaboeacaqGobaacaGLOaGaay zkaaWaaeWaaeaacaqGnbGaaeOtaaGaayjkaiaawMcaaaaa@5EDB@

Δ= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOKH4Qaey iLdqKaeyypa0daaa@3A50@

| 1 2 ( y 2  + y 1 )( x 1  - x 2 ) +  1 2 ( y 1  + y 3 )( x 3  - x 1 ) -  1 2 ( y 2  + y 3 )( x 3  - x 2 ) | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaada WcaaqaaiaabgdaaeaacaqGYaaaamaabmaabaGaaeyEamaaBaaaleaa caqGYaaabeaakiaabccacaqGRaGaaeiiaiaabMhadaWgaaWcbaGaae ymaaqabaaakiaawIcacaGLPaaadaqadaqaaiaabIhadaWgaaWcbaGa aeymaaqabaGccaqGGaGaaeylaiaabccacaqG4bWaaSbaaSqaaiaabk daaeqaaaGccaGLOaGaayzkaaGaaeiiaiaabUcacaqGGaWaaSaaaeaa caqGXaaabaGaaeOmaaaadaqadaqaaiaabMhadaWgaaWcbaGaaeymaa qabaGccaqGGaGaae4kaiaabccacaqG5bWaaSbaaSqaaiaabodaaeqa aaGccaGLOaGaayzkaaWaaeWaaeaacaqG4bWaaSbaaSqaaiaabodaae qaaOGaaeiiaiaab2cacaqGGaGaaeiEamaaBaaaleaacaqGXaaabeaa aOGaayjkaiaawMcaaiaabccacaqGTaGaaeiiamaalaaabaGaaeymaa qaaiaabkdaaaWaaeWaaeaacaqG5bWaaSbaaSqaaiaabkdaaeqaaOGa aeiiaiaabUcacaqGGaGaaeyEamaaBaaaleaacaqGZaaabeaaaOGaay jkaiaawMcaamaabmaabaGaaeiEamaaBaaaleaacaqGZaaabeaakiaa bccacaqGTaGaaeiiaiaabIhadaWgaaWcbaGaaeOmaaqabaaakiaawI cacaGLPaaaaiaawEa7caGLiWoaaaa@6D2F@

Δ= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOKH4Qaey iLdqKaeyypa0daaa@3A50@

| 1 2 { x 1 ( y 2  + y 1  - y 1  - y 3 )  + x 2 ( -y 2  - y 1  + y 2  + y 3 )  + x 3 ( y 1  + y 3  - y 2  - y 3 ) } | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaada WcaaqaaiaabgdaaeaacaqGYaaaamaacmaabaGaaeiEamaaBaaaleaa caqGXaaabeaakmaabmaabaGaaeyEamaaBaaaleaacaqGYaaabeaaki aabccacaqGRaGaaeiiaiaabMhadaWgaaWcbaGaaeymaaqabaGccaqG GaGaaeylaiaabccacaqG5bWaaSbaaSqaaiaabgdaaeqaaOGaaeiiai aab2cacaqGGaGaaeyEamaaBaaaleaacaqGZaaabeaaaOGaayjkaiaa wMcaaiaabccacaqGRaGaaeiiaiaabIhadaWgaaWcbaGaaeOmaaqaba Gcdaqadaqaaiaab2cacaqG5bWaaSbaaSqaaiaabkdaaeqaaOGaaeii aiaab2cacaqGGaGaaeyEamaaBaaaleaacaqGXaaabeaakiaabccaca qGRaGaaeiiaiaabMhadaWgaaWcbaGaaeOmaaqabaGccaqGGaGaae4k aiaabccacaqG5bWaaSbaaSqaaiaabodaaeqaaaGccaGLOaGaayzkaa GaaeiiaiaabUcacaqGGaGaaeiEamaaBaaaleaacaqGZaaabeaakmaa bmaabaGaaeyEamaaBaaaleaacaqGXaaabeaakiaabccacaqGRaGaae iiaiaabMhadaWgaaWcbaGaae4maaqabaGccaqGGaGaaeylaiaabcca caqG5bWaaSbaaSqaaiaabkdaaeqaaOGaaeiiaiaab2cacaqGGaGaae yEamaaBaaaleaacaqGZaaabeaaaOGaayjkaiaawMcaaaGaay5Eaiaa w2haaaGaay5bSlaawIa7aaaa@7419@

Δ =  1 2 | x 1 ( y 2  - y 3 )  + x 2 ( y 3  - y 1 )  + x 3 ( y 1  - y 2 ) | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOKH4Qaae iLdiaabccacaqG9aGaaeiiamaalaaabaGaaeymaaqaaiaabkdaaaWa aqWaaeaacaqG4bWaaSbaaSqaaiaabgdaaeqaaOWaaeWaaeaacaqG5b WaaSbaaSqaaiaabkdaaeqaaOGaaeiiaiaab2cacaqGGaGaaeyEamaa BaaaleaacaqGZaaabeaaaOGaayjkaiaawMcaaiaabccacaqGRaGaae iiaiaabIhadaWgaaWcbaGaaeOmaaqabaGcdaqadaqaaiaabMhadaWg aaWcbaGaae4maaqabaGccaqGGaGaaeylaiaabccacaqG5bWaaSbaaS qaaiaabgdaaeqaaaGccaGLOaGaayzkaaGaaeiiaiaabUcacaqGGaGa aeiEamaaBaaaleaacaqGZaaabeaakmaabmaabaGaaeyEamaaBaaale aacaqGXaaabeaakiaabccacaqGTaGaaeiiaiaabMhadaWgaaWcbaGa aeOmaaqabaaakiaawIcacaGLPaaaaiaawEa7caGLiWoaaaa@5F1F@

Start your learning Journey !

Tutormate-googleplay
Tutormate-AppleStore
Get SMS link to download the app