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.
Solution:
Proof:
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Statement |
Reason |
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1. In ∆MXY, MX = MY. |
1. Tangents drawn to a circle from an external point are equal. |
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2. ∠MXY = ∠MYX = x°. |
2. From statement 1. |
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3. ∠XMY = 180° - x°. |
3.Sum of three angles of a triangle is 180°. |
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4. OX ⊥ XM, i.e., ∠OXM = 90°. |
4. Radius ⊥ tangent. |
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5. ∠OXY = 90° - ∠MXY ⟹ ∠OXY = 90° - x° ⟹ 2∠OXY = 180° - 2x° |
5. From the figure. |
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6. Therefore, ∠XMY = 2∠OXY. (Proved). |
6. From statements 3 and 5. |
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