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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 - Geometry

Sum

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

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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

Concept: Tangent to a Circle
  Is there an error in this question or solution?
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