Equilateral Triangle Inscribed

We will prove that, PQR is an equilateral triangle inscribed in a circle. The tangents at P, Q and R form the triangle P’Q’R’. Prove that P’Q’R’ is also an equilateral triangle.

Equilateral Triangle Inscribed

Solution:

Given:

PQR is an equilateral triangle inscribed in a circle whose centre is O. Three tangents are drawn to the circle at the points P, Q and R. They intersect pair wise at P’, Q’ and R’.

To prove: P’Q’R’ is an equilateral triangle.

Proof:

Statement

Reason

1. ∠Q’PR = ∠PQR.

∠Q’RP = ∠PQR.

1. Angle between tangent and chord is equal to the angle in the alternate segment.

2. ∠PQR = 60°

2. Each angle of the equilateral triangle PQR has the measure 60°.

3. ∠Q’PR = ∠Q’RP = 60°.

3. By statements 1 and 2.

4. ∠PQ’R = 180° – (∠Q’PR + Q’RP)

             = 180° – (60° + 60°) = 60°.

4. Sum of three angles of a triangle is 180.

5. Similarly, ∠PR’Q = 60°

                and ∠QP’R = 60°.

5. Similar reason.

6. ∆P’Q’R’ is equilateral.

6. By statements 4 and 5.





10th Grade Math

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