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Chapter 9: Q14. (page 336)

Quad. ABCD is circumscribed about a circle. Discover and prove a relationship between AB+DC and AD+BC.

Short Answer

Expert verified

AB+DCand AD+BC

Step by step solution

01

Step 1. Given information.

Let ABCD be a quadrilateral circumscribing the circle with centre O. The quadrilateral touches the circle at point P,Q,R and S.

02

Step 2. Formula used.

Lengths of tangents drawn from external point are equal.

03

Step 3. Proof.

Consider the figure below,

According to theorem, lengths of tangents drawn from external point are equal.

Then,

AP=ASBP=BQCR=CQDR=DS

04

Step 4. Adding above equations.

AP+BP+CR+DR=AS+BQ+CQ+DSAP+BP+CR+DR=AS+SD+CQ+BQAB+CD鈥夆赌�=鈥�AD+BC

Hence, it is proved that, AB+CD=AD+BC

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For each exercise draw 鈯�O with radius 12. Then draw radii OA炉 and to OB炉 form an angle with the measure named. Find the length of AB炉.

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