A quadrilateral is drawn to circumscribe a circle (see Fig. 10.12). Prove that

Understand the Question A quadrilateral circumscribes a circle, meaning all four of its sides are tangent to the circle at points and . The lengths of two tangents drawn from an external point to a circle are equal. Treating each vertex () as an external point gives four pairs of equal tangent segments, which can be added and grouped to prove . Step 1 · Write Equations for Tangents from Each Vertex Since the lengths of tangents drawn from an external point to a circle are equal: From external points and : Step 2 · Add Equations and Group Terms Adding equations and : Grouping terms to form the complete sides of the quadrilateral: Substituting the side lengths: Answer > Hence proved, . Common Mistakes Mismatched Sides: Writing equations with terms on the wrong side (e.g., writing instead of ). Keep on the same side so they add up directly to . Incorrect Grouping: Grouping non-adjacent segments (e.g., trying to add instead of adjacent collinear segments ). Each pair must lie on the same side of the quadrilateral.

Circles | Exercise 10.2

Question 8

A quadrilateral $ABCD$ is drawn to circumscribe a circle (see Fig. 10.12). Prove that

$\text{AB} + \text{CD} = \text{AD} + \text{BC}$

Solution

Understand the Question

A quadrilateral $ABCD$ circumscribes a circle, meaning all four of its sides are tangent to the circle at points $P, Q, R,$ and $S$ .
The lengths of two tangents drawn from an external point to a circle are equal.
Treating each vertex ( $A, B, C, D$ ) as an external point gives four pairs of equal tangent segments, which can be added and grouped to prove $\text{AB} + \text{CD} = \text{AD} + \text{BC}$ .

Step 1 · Write Equations for Tangents from Each Vertex

Since the lengths of tangents drawn from an external point to a circle are equal: Diagram 1

From external points $A, B, C,$ and $D$ :

\begin{aligned} \text{AP} &= \text{AS} \quad \dots (1) \\[0.6em] \text{BP} &= \text{BQ} \quad \dots (2) \\[0.6em] \text{CR} &= \text{CQ} \quad \dots (3) \\[0.6em] \text{DR} &= \text{DS} \quad \dots (4) \end{aligned}

Step 2 · Add Equations and Group Terms

Adding equations $(1), (2), (3),$ and $(4)$ : $\text{AP} + \text{BP} + \text{CR} + \text{DR} = \text{AS} + \text{BQ} + \text{CQ} + \text{DS}$

Grouping terms to form the complete sides of the quadrilateral: $(\text{AP} + \text{BP}) + (\text{CR} + \text{DR}) = (\text{AS} + \text{DS}) + (\text{BQ} + \text{CQ})$

Substituting the side lengths: $\text{AB} + \text{CD} = \text{AD} + \text{BC}$

Answer

Hence proved, $\text{AB} + \text{CD} = \text{AD} + \text{BC}$ .

Common Mistakes

Mismatched Sides: Writing equations with terms on the wrong side (e.g., writing $\text{BQ} = \text{BP}$ instead of $\text{BP} = \text{BQ}$ ). Keep $\text{AP}, \text{BP}, \text{CR}, \text{DR}$ on the same side so they add up directly to $\text{AB} + \text{CD}$ .
Incorrect Grouping: Grouping non-adjacent segments (e.g., trying to add $\text{AP} + \text{CR}$ instead of adjacent collinear segments $\text{AP} + \text{BP}$ ). Each pair must lie on the same side of the quadrilateral.

Question 8

More questions in Exercise 10.2