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A quadrilateral ABCD is drawn to circumscribe a circle, as shown in the figure. Prove that AB + CD = AD + BC - Mathematics

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Sum

A quadrilateral ABCD is drawn to circumscribe a circle, as shown in the figure. Prove that AB + CD = AD + BC

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Solution

We known that the lengths of tangents drawn from an exterior point to a circle are equal.

∴ AP = AS ….(i) [tangents from A]

BP = BQ ….(ii) [tangents from B]

CR = CQ ….(iii) [tangents from C]

DR = DS …(iv) [tangents from D]

∴ AB + CD = (AP + BP) + (CR + DR)

= (AS + BQ) + (CQ + DS) [using (i), (ii), (iii), (iv)]

= (AS + DS) + (BQ + CQ)

= (AD + BC).

Hence, (AB + CD) = (AD + BC)

Concept: Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
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