Exercise 4.6 - Chapter 4 Geometry 9th Maths Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

Ex $4.6$
Question $1 .$
Draw a triangle $\mathrm{ABC}$, where $\mathrm{AB}=8 \mathrm{~cm}, \mathrm{BC}=6 \mathrm{~cm}$ and $\angle \mathrm{B}=70^{\circ}$ and locate its circumcentre and draw the circumcircle.
Solution:
$\triangle \mathrm{ABC}$, where $\mathrm{AB}=8 \mathrm{~cm}$,
$\begin{aligned}
&\mathrm{BC}=6 \mathrm{~cm} \\
&\mathrm{~B}=70^{\circ}
\end{aligned}$

Construction:
(i) Draw the $\triangle \mathrm{ABC}$ with the given measurements.
(ii) Construct the perpendicular bisector at any two sides $(\mathrm{AB}$ and $\mathrm{BC})$ and let them meet at $\mathrm{S}$ which is the circumcircle.
(iii) $\mathrm{S}$ as centre and $\mathrm{SA}=\mathrm{SB}=\mathrm{SC}$ as radius, draw the circumcircle to pass through $\mathrm{A}, \mathrm{B}$, and $\mathrm{C}$. Circum radius $=4.3 \mathrm{~cm}$.
 

Question $2 .$
Construct the right triangle $\mathrm{PQR}$ whose perpendicular sides are $4.5 \mathrm{~cm}$ and $6 \mathrm{~cm}$. Also locate its circumcentre and draw the circumcircle.
Solution:
Right triangle $P Q R$ whose perpendicular sides are $4.5 \mathrm{~cm}$ and $6 \mathrm{~cm}$

Construction :
(i) Draw the right triangle $\mathrm{PQR}$ with the given measurements.
(ii) Construct the perpendicular bisector at any two sides ( $\mathrm{PQ}$ and $\mathrm{QR})$ and let them meet at $\mathrm{S}$ which is the circumcentre.
(iii) $\mathrm{S}$ as centre and $\mathrm{SP}=\mathrm{SQ}=\mathrm{SR}$ as radius, draw the circumcircle to pass through $\mathrm{P}, \mathrm{Q}$ and $\mathrm{R}$. Circumradius $=3.7 \mathrm{~cm}$.

 Question $3 .$
Construct $\triangle \mathrm{ABC}$ with $\mathrm{AB}=5 \mathrm{~cm} \angle \mathrm{B}=100^{\circ}$ and $\mathrm{BC}=6 \mathrm{~cm}$. Also locate its circumcentre draw circumcircle.
Solution:

Construction :
(i) Draw the $\triangle \mathrm{ABC}$ with the given measurements.
(ii) Construct the perpendicular bisector at any two sides (BC and $\mathrm{AC}$ ) and let them meet at $\mathrm{S}$ which is the circumcentre.
(iii) $\mathrm{S}$ as centre and $\mathrm{SA}=\mathrm{SB}=\mathrm{SC}$ as radius, draw the circumcircle to pass through $\mathrm{A}, \mathrm{B}$, and $\mathrm{C}$.
Circumradius $=4.3 \mathrm{~cm}$.
 

Question $4 .$
Construct an isosceles triangle $\mathrm{PQR}$ where $\mathrm{PQ}=\mathrm{PR}$ and $\angle \mathrm{Q}=50^{\circ}, \mathrm{QR}=7 \mathrm{~cm}$. Also draw its circumcircle.
Solution:
Isosceles triangle $\mathrm{PQR}$ where $\mathrm{PQ}=\mathrm{PR}$ and $\mathrm{Q}=50^{\circ}, \mathrm{QR}=7 \mathrm{~cm}$.

Construction :
(i) Draw the $\triangle \mathrm{PQR}$ with the given measurements.
(ii) Construct the perpendicular bisector at any two sides (PQ and $\mathrm{QR}$ ) and let them meet at $\mathrm{S}$ which is the circumcentre.
(iii) $\mathrm{S}$ as centre and $\mathrm{SP}=\mathrm{SQ}=\mathrm{SR}$ as radius, draw the circumcircle to pass through $\mathrm{P}, \mathrm{Q}, \mathrm{R}$. Circumradius $=$ $3.5 \mathrm{~cm}$.
 

Question $5 .$
Draw an equilateral triangle of sides $6.5 \mathrm{~cm}$ and locate its incentre. Also draw the incircle.
Solution:
Side $=6.5 \mathrm{~cm}$

Construction :
Step 1 : Draw $\triangle \mathrm{ABC}$ with $\mathrm{AB}=\mathrm{BC}=\mathrm{CA}=6.5 \mathrm{~cm}$
Step 2 : Construct angle bisectors of any two angles (A and B) and let them meet at I.I is the incentre of $\triangle \mathrm{ABC}$.
Step 3 : Draw perpendicular from I to any one of the side $(\mathrm{AB})$ to meet $\mathrm{AB}$ at $\mathrm{D}$.
Step 4 : With I as centre, ID as radius draw the circle. This circle touches all the sides of triangle internally.
Step $5:$ Measure in radius. In radius $=1.9 \mathrm{~cm}$.
 

Question $6 .$
Draw a right triangle whose hypotenuse is $10 \mathrm{~cm}$ and one of the legs is $8 \mathrm{~cm}$. Locate its incentre and also draw the incircle.
Solution:
hypotenuse $=10 \mathrm{~cm}$
One of the legs $=8 \mathrm{~cm}$

Step $1:$ Draw $\mathrm{AABC}$ with $\mathrm{BC}=8 \mathrm{~cm} . \mathrm{AC}=10 \mathrm{~cm}$ with right angle at $\mathrm{B}$.
Step 2 : Construct angle bisectors of any two angles (B and $\mathrm{C}$ ) and let them meet at 1 .l is the incentre.
Step 3 : Draw perpendicular from I to any side of the triangle to meet BC at D.
Step 4 : With I as centre, ID as radius draw the incircle, which touches all the three sides of the triangle internally. In radius $=1.9 \mathrm{~cm}$.

Question 7.
Draw $\triangle \mathrm{ABC}$ given $\mathrm{AB}=9 \mathrm{~cm}, \angle \mathrm{CAB}=115^{\circ}$ and $\triangle \mathrm{ABC}=40^{\circ}$. Locate its incentre and also draw the incircle. (Note: You can check from the above examples that the incentre of any triangle is always in its interior).
Solution:

Construction :
Step 1: Draw $\triangle \mathrm{ABC}$ with $\mathrm{AB}=9 \mathrm{~cm} . \angle \mathrm{A}=115^{\circ}, \angle \mathrm{B}=40^{\circ}$.
Step 2 : Construct angle bisectors of any two angles $(B$ and $C$ ). Let them meet at I.I is the incentre of $\triangle \mathrm{ABC}$.
Step 3 : Draw perpendicular from I to any side $(B C)$ to meet $B C$ at D.
Step 4 : Draw incircle, with I as centre and ID a radius. Measures the in radius.
 

Question 8.
Construct $\triangle \mathrm{ABC}$ in which $\mathrm{AB}=\mathrm{BC}=6 \mathrm{~cm}$ and $\mathrm{B}=80^{\circ}$. Locate its incentre and draw the incircle.
Solution:
In $\triangle \mathrm{ABC}, \mathrm{AB}=\mathrm{BC}=6 \mathrm{~cm}, \angle \mathrm{B}=80^{\circ}$.

Construction :
Step 1 : Draw $\mathrm{AABC}$ with $\mathrm{BC}=6 \mathrm{~cm} . \mathrm{AB}=6 \mathrm{~cm}, \mathrm{AB}=6 \mathrm{~cm}$, and $\angle \mathrm{B}=80^{\circ}$.
Step 2 : Construct the incentre I and ID is the in radius, as in the previous sums.
Step 3 : Draw incircle with I as centre and ID as radius. It touches all the three sides internally.
Step 4 : Measure in radius. In radius $=1.7$