Geometry

Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

$Math Studyset 91影视 Explanations$ Math

Student Edition 路 227 pages

ISBN: 9780395977279

Chapter 8: Q44. (page 289)

Given: $P Q R S$ is a rectangle.
$鈭� A$ is a right angle.
Prove: $B S 路 R C = P S 路 Q R = {(P S)}^{2}$

Short Answer

Expert verified

The result is,

$B S 路 R C = P S 路 Q R = P S^{2}$ .

Step by step solution

Step 1. Given information.

The triangle is shown below:

$P Q R S$ is rectangle.

Step 2. Consider triangles BSP and PAQ.

$\begin{array}{l} 鈭� B S P = 鈭� P A Q = 90 掳 \\ 鈭� P B S = 鈭� Q P A 鈥勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌� (Correspondingangle) \\ 鈭� 螖 B S P ~ 螖 P A Q 鈥勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌� ...... (1) \end{array}$

Step 3. Consider triangles PAQ and QRC.

$\begin{array}{l} 鈭� Q R C = 鈭� P A Q = 90 掳 \\ 鈭� P Q A = 鈭� Q C R 鈥勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌� (Correspondingangle) \\ 鈭� 螖 Q R C ~ 螖 P A Q 鈥勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌� ...... (2) 鈥� \end{array}$

From equation $(1)$ and $(2)$ ,

$螖 Q R C ~ 螖 B S P$

Hence,

$\begin{array}{l} \frac{B S}{Q R} = \frac{P S}{R C} \\ B S 鈰� R C = P S 鈰� Q R 鈥勨赌勨赌勨赌� ...... 鈥� (3) \end{array}$

Also, $P Q R S$ is rectangle.

Hence, side $P S$ is equal to $Q R$ .

$\begin{matrix} B S 鈰� R C = P S 鈰� P S \\ B S 鈰� R C = P S^{2} 鈥勨赌勨赌勨赌� ...... 鈥� (4) \end{matrix}$

Thus, from equations $(3)$ and $(4)$ ,

$B S 路 R C = P S 路 Q R = P S^{2}$

Therefore, it is proved that, $B S 路 R C = P S 路 Q R = P S^{2}$ .

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91影视

Given: $P Q R S$ is a rectangle.
$鈭� A$ is a right angle.
Prove: $B S 路 R C = P S 路 Q R = {(P S)}^{2}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Consider triangles BSP and PAQ.

Step 3. Consider triangles PAQ and QRC.

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91影视

Given: PQRSis a rectangle. 鈭�A is a right angle.Prove: BS路RC=PS路QR=PS2

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Consider triangles BSP and PAQ.

Step 3. Consider triangles PAQ and QRC.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Discrete Mathematics

Logic and Functions

Geometry

Probability and Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.

Given: $P Q R S$ is a rectangle.
$鈭� A$ is a right angle.
Prove: $B S 路 R C = P S 路 Q R = {(P S)}^{2}$