Circles and Geometric Shapes | Exercise 5.2

Question 1

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

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Solution

We will show that two sides of the triangle are radii of the circle.

Step 1 — Setting up the triangle

Let's consider a circle. Let its center be O. Let AB be a chord of this circle. We need to form a triangle. Let's join point O to point A. Let's also join point O to point B. This forms triangle OAB.

Diagram 1

Step 2 — Proving it is isosceles

Look at side OA in triangle OAB. OA is a radius of the circle. Now look at side OB. OB is also a radius of the circle. All radii of the same circle are equal. So, we can say that OA = OB. A triangle with two equal sides is isosceles. Therefore, triangle OAB is an isosceles triangle. This proves the statement.

Answer

The triangle formed by a chord and the centre of the circle is isosceles because the two sides connecting the centre to the endpoints of the chord are radii of the same circle, and all radii are equal in length.

More questions in Exercise 5.2

Q1

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

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