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# Constructions

## 11 CONSTRUCTIONS

Division of Line segment:

Construction 1: To divide line segment of $\text{6}$ cm in a given ratio $\text{3:2}$.

Steps :

1)   Draw a line segment $\text{AB}$ of $\text{6}$ cm.

2) Draw ray $\text{AX}$ making acute angle with $\text{AB}$

3) Draw an arc of any measure , taking $\text{A}$ as centre to cut the ray $\text{AX}$ in $\text{A1}$.

4) Draw arc taking centre at $\text{A1}$ and measure $\text{AA1}$. Locate point $\text{A2}$ on $\text{AX}$ as step $\text{3}$.

5) Repeat the step as in and locate points .

We locate $\text{5}$ points as ratio is $\text{3:2}$ and .

6) Join $\text{BA5}$

7) Draw a line parallel to $\text{BA5}$ from $\text{A3}$

8) Let this line cuts $\text{AB}$ at $\text{C}$

9 ) The point $\text{C}$ is required point.

Construction 2 :To divide line segment of $\text{6}$ cm in a given ratio $\text{5:4}$

1)   Draw a line segment $\text{AB}$ of $\text{6}$ cm.

2) Draw ray $\text{AX}$,and $\text{BY}$ making acute angle with $\text{AB}$

3) Draw an arc of any measure , taking $\text{A}$ as centre, to cut the ray $\text{AX}$ in $\text{A1}$.

3a) Draw an arc of any measure , taking $\text{B}$ as centre, to cut the ray $\text{BY}$ in $\text{B1}$.

4) Draw an arc taking centre at $\text{A1}$ and measure $\text{AA1}$. Locate point $\text{A2}$ on $\text{AX}$ as step $\text{3}$.

4a) Draw an arc taking centre at $\text{B1}$ and measure $\text{AA1}$. Locate point $\text{B2}$ on $\text{BY}$ as step $\text{3}$.

5) Repeat the step as in $\text{3a}$ and cand locate points .

6) Join $\text{A5}$ and $\text{B4}$.

7) Let it cut the segment $\text{AB}$ in point $\text{C}$.

8) ThepointC is required point with ratio

5:4.

Construct a triangle similar to given triangle as per given scale factor.

Construct a triangle similar to triangle with $\Delta \text{ABC}$

with sides equal to of corresponding

sides of $\Delta \text{ABC}$

1)   Draw a triangle $\text{ABC}$ of given measure with base as $\text{BC}$.

2) Draw a ray $\text{BX}$ making acute angle with $\text{BC}$.

3)Draw $\text{5}$ equaidistant arcs to locate points .( 5 points as ratio of sides is and

largerof it is $\text{5}$.

Incomplete figure

4) Join $\text{A5}$ and $\text{C}$.

5) Draw a line parallel to $\text{A5C}$ from $\text{A3}$ and let it cut $\text{BC}$ at $\text{p}$.

6) Draw a line parallel to side $\text{AC}$ from $\text{p}$.

7) Let this line cuts side $\text{AB}$ at $\text{Q}$.

8) SO $\Delta \text{BPO}$ is required triangle.

Construction of a tangent to a circle at a given point on the circle.

1)Construction of a tangent to a circle at a given point on the circle. (when centre is known

0

1) Draw a circle with centre $\text{O}$.

2) Mark a point $\text{p}$ on the circle.

3) Join $\text{PO}$

4) construct angle $\text{90°}$ at P.

2)

Construction of a tangent to a circle at a given point on the circle. (when centre is unknown

1)   trace a circle.

2)Draw a chord $\text{PQ}$.

3) Take a point $\text{R}$ on circle

4) Join $\text{PR}$ and $\text{QR}$.

5) Constuct $\angle \text{QPY}$ equal to $\angle \text{PRQ}$ and on opposite side of chord $\text{PQ}$

6) Line $\text{PY}$ is required tangent.

Image required

Construction of a tangent to a circle from a point outside the circle.

1) Draw a tangent to a circle from a point outside it.(when centre is known)

1)   Draw a circle with centre $\text{O}$.

2) Mark a point $\text{P}$ outside a circle.

3) Join $\text{PO}$

4) Draw perpendicular bisector of $\text{OP}$

5) Let it meet $\text{OP}$ at $\text{M}$.

6) Draw a circle with $\text{M}$ as centre and $\text{MP}$ as radius.

7) Let the circle cuts given circle at $\text{Q}$ and $\text{R}$.

8) Join $\text{PQ}$ and $\text{PR}$.

9) $\text{PQ}$ and $\text{PR}$ are the required tangents to the given circle.

1) Draw a tangent to a circle from a point outside it.(when centre is unknown)

1)               Take $\text{P}$, a point outside the given circle.

2) Through $\text{P}$ draw a secant $\text{AB}$ to intersect the circle at $\text{A}$ and $\text{B}$ (say).

Consult while drawing figure

3. Take point C such that i.e., P is the mid-point of $\text{AC}$.’

4. Draw a semi-circle with $\text{BC}$ as diameter.

Incomplete fig.

5. Draw $\text{PD}\perp \text{CB}$, intersecting the semi-circle at $\text{D}$.

6. With $\text{PD}$ as radius and $\text{P}$ as centre, draw arcs to intersect the given circle at $\text{T}$ and $\text{T}$ ’.

7. Join $\text{PT}$ and $\text{PT}$ ’, $\text{PT}$ and $\text{PT}$ ’ are the required tangents.