Two Tangents are Drawn to a Circle from an External Point

We will prove that the tangents MX and MY are drawn to a circle with centre O from an external point M. Prove that ∠XMY = 2∠OXY.

Two Tangents are Drawn to a Circle from an External Point

Solution:

Proof:

Statement

Reason

1. In ∆MXY, MX = MY.

1. Tangents drawn to a circle from an external point are equal.

2. ∠MXY = ∠MYX = x°.

2. From statement 1.

3. ∠XMY = 180° - x°.

3.Sum of three angles of a triangle is 180°.

4. OX ⊥ XM, i.e., ∠OXM = 90°.

4. Radius ⊥ tangent.

5. ∠OXY = 90° - ∠MXY

⟹ ∠OXY = 90° - x°

⟹ 2∠OXY = 180° - 2x°

5. From the figure.

6. Therefore, ∠XMY = 2∠OXY. (Proved).

6. From statements 3 and 5.






10th Grade Math

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