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Question
Prove that the lengths of two tangent segments drawn to the circle from an external point are equal.
Solution
Given: O is the centre of the circle and P is a point in the exterior of the circle. A and B are the points of contact of the two tangents from P to the circle.
To Prove: PA = PB
Construction: Draw seg OA, seg OB, and seg OP.
Proof: Line AP ⊥ radius OA and line BP ⊥ radius OB ... (Tangent perpendicular to radius)
∴ `angle"PAO" = angle"PBO" = 90^@`
In right-angled triangles `triangle "OAP"` and `triangle"OBP"`
hypotenuse OP ≅ hypotenuse OP ...(Common side)
seg OA ≅ seg OB ...(Radii of the same circle)
`:.triangle"OAP" ≅ triangle"OBP"` ...(Hypotenuse-side of theorem)
∴ seg PA ≅ seg PB ...(c.s.c.t.)
∴ PA = PB
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