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⭕ Chapter 5: I'm Up and Down, and Round and Round • Exercise Set 5.2

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Exercise Set 5.1 3Think, Draw and Infer 1Exercise Set 5.2 2Exercise Set 5.3 3Exercise Set 5.4 2Exercise Set 5.5 3Exercise Set 5.6 1End-of-Chapter Exercises 12All Exercises 27
Q1

Show that the triangle formed by a chord and the centre of the circle is isosceles.

Answer: Let the circle have centre $C$ and let the chord be $AB$. Form $\Delta CAB$ by joining the centre $C$ to the endpoints of the chord $A$ and $B$. Both segments $CA$ and $CB$ connect the centre to points on the circle, so they are both radii of the circle. Since all radii of a given circle are equal in length, $CA = CB$. A triangle with two equal sides is an isosceles triangle.

C A B r r
CA = CB = r (radii of the circle), so ΔCAB is isosceles
Q2

Show that if two such isosceles triangles (occurring in the previous question) have equal base length, they are congruent to each other.

Answer: Let the two triangles be $\Delta CAB$ and $\Delta CDE$, where $C$ is the centre of the circle, and $AB$ and $DE$ are two chords (the bases of the triangles).
1. $CA = CD$ (both are radii of the circle).
2. $CB = CE$ (both are radii of the circle).
3. $AB = DE$ (given that the bases are of equal length).
By the Side-Side-Side (SSS) congruence criterion, $\Delta CAB \cong \Delta CDE$.

C D E A B
By SSS congruence: CA = CD = r, CB = CE = r, and AB = DE ⇒ ΔCAB ≅ ΔCDE