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Question
In the following figure, seg AB is the diameter of the circle with center P. Line CB be the tangent and line AC intersects a circle in point D. Prove that:
AC x AD = 4 (radius)2
Solution
Given: A circle with center P. CB tangent and line AC intersect a circle in point D
Construction: Join BD.
To Prove: ∴ ∠ADB =90° [Angle inscribed in semicircle]
∴ ∠PBC = 90° [Tangent perpendicular to the radius]
i.e. ∠ABC =90°
In Δ ACB and Δ ABD
∠ ABC = ∠ ADB [Each is of 90°]
∴ ∠ CAB = ∠DAB [Common angle]
∴ ΔACB ∼ ΔABD [AA property]
∴ `"AC"/"AB"="AB"/"AD"`
∴AC × AD = (AB)2…(1)
AP = PB …(radii of the same circle)
∴ AB = AP +PB
∴ AB = 2AP
Substituting the value of AB in equation (1)
AC × AD = (2AP)2
∴ AC × AD = 4(AP)2
∴ AC × AD =4 (radius)2
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