NCERT Solutions for Class 9 Maths Chapter 5 — Exercise Set 5.2#
Theorem 2: Equal chords of a circle subtend equal angles at the centre.
Theorem 3: Chords of a circle that subtend equal angles at the centre are equal.
All radii of a circle are equal — this fact is used in almost every proof in this chapter.
1. Show that the triangle formed by a chord and the centre of the circle is isosceles. #
Solution
Let $AB$ be a chord of a circle with centre $C$ and radius $r$. Joining $C$ to $A$ and $B$ gives $\triangle CAB$.
$CA$ and $CB$ are both radii of the circle:
$$CA = CB = r$$A triangle with two equal sides is isosceles.
∴ $\triangle CAB$ is isosceles, with $AB$ as its base. (Its base angles are also equal: $\angle CAB = \angle CBA$.)
2. Show that if two such isosceles triangles (occurring in the previous question) have equal base length, they are congruent to each other. #
Solution
Let $AB$ and $DE$ be two chords of the same circle, with centre $C$ and radius $r$, such that $AB = DE$. Consider $\triangle CAB$ and $\triangle CDE$.
| Part | Reason |
|---|---|
| $CA = CD$ | both are radii, equal to $r$ |
| $CB = CE$ | both are radii, equal to $r$ |
| $AB = DE$ | given (equal bases) |
All three sides of one triangle are equal to the three sides of the other.
∴ By SSS congruence, $\triangle CAB \cong \triangle CDE$.
(The two chords must be in the same circle, or in circles of equal radius; otherwise the equal sides $CA$ and $CD$ would not match.)