Question 17 of 50intermediate🔧 ApplyShort Answer2 marks

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

Correct Answer

Consider two isosceles triangles formed by chords and the centre of a circle, say ΔCAB\Delta CAB and ΔCDE\Delta CDE. Since the sides CA, CB, CD, and CE are radii of the circle, we have CA = CB and CD = CE. Therefore, CA = CD and CB = CE.

We are given that the base lengths of these two isosceles triangles are equal, so AB = DE. Since all three sides of ΔCAB\Delta CAB are equal to the corresponding three sides of ΔCDE\Delta CDE (CA = CD, CB = CE, and AB = DE), the triangles are congruent by the SSS congruence rule. Hence, ΔCAB≅ΔCDE\Delta CAB \cong \Delta CDE.

Exercise: Exercise Set 5.2 | Q: 2 | (Page 9)
For More Understanding

Explanation

The textbook context establishes that the triangle formed by a chord and the centre is isosceles because its two sides are radii of the circle. When comparing two such isosceles triangles with equal base lengths (chords), their corresponding sides from the centre are equal (being radii). Applying the Side-Side-Side (SSS) congruence criterion proves the triangles are congruent.

Solution Steps

  1. Step 1: Identify that the sides from the centre to the chords are radii, so CA = CB and CD = CE, which means CA = CD and CB = CE.

  2. Step 2: Use the given condition that the base lengths are equal (AB = DE).

  3. Step 3: Apply the SSS congruence rule (CA = CD, CB = CE, AB = DE) to conclude that the two isosceles triangles are congruent.

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