Chapter 9: Q14. (page 336)

Quad. $A B C D$ is circumscribed about a circle. Discover and prove a relationship between $A B + D C$ and $A D + B C .$

Short Answer

Expert verified

$A B + D C$ and $A D + B C$

Step by step solution

Step 1. Given information.

Let $A B C D$ be a quadrilateral circumscribing the circle with centre $O$ . The quadrilateral touches the circle at point $P, Q, R$ and $S$ .

Step 2. Formula used.

Lengths of tangents drawn from external point are equal.

Step 3. Proof.

Consider the figure below,

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

Then,

$\begin{array}{l} A P = A S \\ B P = B Q \\ C R = C Q \\ D R = D S \end{array}$

Step 4. Adding above equations.

$\begin{array}{l} A P + B P + C R + D R = A S + B Q + C Q + D S \\ (A P + B P) + (C R + D R) = (A S + S D) + C Q + B Q \\ A B + C D = A D + B C \end{array}$

Hence, it is proved that, $A B + C D = A D + B C$

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Quad. $A B C D$ is circumscribed about a circle. Discover and prove a relationship between $A B + D C$ and $A D + B C .$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Formula used.

Step 3. Proof.

Step 4. Adding above equations.

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Formula used.

Step 3. Proof.

Step 4. Adding above equations.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Pure Maths

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Probability and Statistics

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

Quad. $A B C D$ is circumscribed about a circle. Discover and prove a relationship between $A B + D C$ and $A D + B C .$