Embibe Experts Solutions for Chapter: Constructions, Exercise 1: Exercise

Arrange the following steps of construction while dividing a line segment length $12 \mathrm{cm}$ internally in the ratio $2:5$.

Steps of construction:

Step $1:$ Locate $7\left(=m+n\right)$ points ${\mathrm{A}}_1, {\mathrm{A}}_2,......., {\mathrm{A}}_7$ on $\mathrm{AX}$ such that ${\mathrm{AA}}_1={\mathrm{A}}_1{\mathrm{A}}_2=...........={\mathrm{A}}_6{\mathrm{A}}_7.$

Step $2:$ Draw a line segment $\mathrm{AB}$ of length $12 \mathrm{cm}$.

Step $3:$ Through ${\mathrm{A}}_2\left(m=2\right)$, draw a line parallel to ${\mathrm{A}}_7\mathrm{B}$ (by making an angle equal to ${\mathrm{AA}}_7\mathrm{B}$) at ${\mathrm{A}}_2$ so as to intersect $\mathrm{AB}$ at the point $\mathrm{C}$.

Step $4:$ Join ${\mathrm{BA}}_7$.

Step $5:$ Draw any ray $\mathrm{AX}$, making an acute angle with $\mathrm{AB}$.

Draw a circle of radius $3 \mathrm{cm}$. Construct a pair of tangents from an external point which is at a distance of $5 c\mathrm{m}$. Measure the lengths of the tangents and verify your measure by geometrical measuring. 

Construct a circle with centre $O$ and radius $2.5 \mathrm{cm}$. Take a point $P$ at a distance $7 \mathrm{cm}$ from $O$. Draw two tangents to the circle from $P$.

Construct a circle with radius $4.5 \mathrm{cm}$ and draw two tangents to the circle from the external point at a distance $7.5 \mathrm{cm}$ from the centre. Find the length of the tangents and find the relation between them.

Construct a pair of tangents from an external point which is at a distance of $5.5 c\mathrm{m}$ from the centre of a circle whose radius is $2.5 c\mathrm{m}$.

Using pair of compasses and ruler, construct $\Delta \mathrm{PQR}$ with sides $\mathrm{PQ}=6.5 \mathrm{cm},\mathrm{QR}=7 \mathrm{cm}\text{ and }\mathrm{PR}=6 \mathrm{cm}$. Then construct $\Delta {\mathrm{P}}^{\text{'}}{\mathrm{QR}}^{\text{'}}$ whose sides are $\frac{4}{3}$ of the corresponding sides of $\Delta \mathrm{PQR}$.

Using a pair of compasses and ruler, construct $∆\mathrm{ABC}$ with sides $\mathrm{AB}=8 \mathrm{cm}$, $\mathrm{BC}=9.5 \mathrm{cm}$, and $AC=6.5 \mathrm{cm}$. Then construct $∆{\mathrm{A}}^{\text{'}}{\mathrm{BC}}^{\text{'}}$ whose sides are $\frac{3}{4}$ of the corresponding sides of $∆\mathrm{ABC}\text{. }$

Draw a circle with centre $O$ and radius $2.7 cm$. Take a point $P$ at a distance $8.5 cm$ from the centre of the circle. Using a pair of compasses and ruler, construct two tangents $PS$ and $PT$ to the circle. Measure and state the length of the tangent segments.