Chapter 26: Chapter 26

Problem 509

Show that in a quadrilateral circumscribed about a circle, the sum of the lengths of a pair of opposite sides equals the sum of the lengths of the remaining pair of opposite sides.

Short Answer

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In a quadrilateral ABCD circumscribed about a circle, let the circle touch the sides AB, BC, CD, and DA at points P, Q, R, and S, respectively. Using tangent properties and the tangent-tangent theorem, we find equalities for the lengths of the segments: AP = AS, BQ = BP, CR = CQ, and DS = DR. Now, summing up the opposite sides, we get (AB + CD) = (x + y + u + v) and (AD + BC) = (s + t + z + w), where x, y, z, w, u, v, s, and t represent the lengths of the segments. Substituting the equal lengths found, we arrive at the conclusion that the sum of the lengths of a pair of opposite sides equals the sum of the lengths of the remaining pair of opposite sides in a quadrilateral circumscribed about a circle.

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TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Draw the quadrilateral circumscribed about a circle

Draw quadrilateral ABCD such that it is circumscribed about a circle. Let the circle touch the sides AB, BC, CD, and DA at points P, Q, R, and S, respectively.

Step 2: Identify the lengths and tangent properties

Let AP = x, PB = y, BQ = z, QC = w, CR = u, RD = v, DS = t, and SA = s. Now, notice that since the circle is tangent to the sides of the quadrilateral, (AP, PS) and (BP, PQ) are both tangent to the circle at point P. (BQ, QR) and (CQ, CP) are both tangent to the circle at point Q. (CR, RD) and (DR, QS) are both tangent to the circle at point R. (DS, SA) and (AS, RP) are both tangent to the circle at point S.

Step 3: Use the Tangent-tangent theorem

Using the tangent-tangent theorem: if two tangents are drawn to a circle from an external point, then their lengths are equal. We find the following equalities: 1) AP = AS -> x = s 2) BQ = BP -> z = y 3) CR = CQ -> u = w 4) DS = DR -> t = v

Step 4: Sum the length of opposite sides

Now, we will find the sum of the lengths of opposite sides of the quadrilateral: 1) AB + CD = AP + PB + CR + RD = x + y + u + v 2) AD + BC = AS + SD + BQ + QC = s + t + z + w

Step 5: Substitute the equal lengths and prove the identity

Substitute the equal lengths found in step 3 into the sums: 1) x + y + u + v = x + z + w + t (substitute z = y and t = v) 2) s + t + z + w = s + u + y + w (substitute u = w and s = x) Now comparing the two sums: (x + y + u + v) = (s + t + z + w) Thus, the sum of the lengths of a pair of opposite sides equals the sum of the lengths of the remaining pair of opposite sides in a quadrilateral circumscribed about a circle.

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Chapter 26: Chapter 26

Show that in a quadrilateral circumscribed about a circle, the sum of the lengths of a pair of opposite sides equals the sum of the lengths of the remaining pair of opposite sides.

Short Answer

Step by step solution

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Step 1: Draw the quadrilateral circumscribed about a circle

Step 2: Identify the lengths and tangent properties

Step 3: Use the Tangent-tangent theorem

Step 4: Sum the length of opposite sides

Step 5: Substitute the equal lengths and prove the identity

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