 Suggested languages for you:

Europe

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.

Expert verified
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.
See the step by step solution

## Unlock all solutions

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.

We value your feedback to improve our textbook solutions.

## Access millions of textbook solutions in one place

• Access over 3 million high quality textbook solutions
• Access our popular flashcard, quiz, mock-exam and notes features ## Join over 22 million students in learning with our Vaia App

The first learning app that truly has everything you need to ace your exams in one place.

• Flashcards & Quizzes
• AI Study Assistant
• Smart Note-Taking
• Mock-Exams
• Study Planner 