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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: Q43. (page 289)

Given: $P Q R S$ is a rectangle.
$P S$ is the geometric mean between $S T$ and $T R$ .
Prove: $鈭� P T Q$ is a right angle.

Short Answer

Expert verified

The angle $P T Q$ is a right angle.

Step by step solution

01

Step 1. Given information.

The rectangle is shown below,

02

Step 2. Concept used.

Side $P S$ is the geometric mean between $S T$ and $T R$ .

Then,

$P S^{2} = S T 脳 T R$

Consider side $P Q$ ,

$P Q = S R (Opposite side of rectangle)$

03

Step 3. Take square both sides.

$\begin{matrix} P Q^{2} = S R^{2} \\ = {(S T + T R)}^{2} \\ = S T^{2} + S T 脳 T R + S T 脳 T R + T R^{2} \\ = (S T^{2} + P S^{2}) + (T R^{2} + P S^{2}) 鈥勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌� (Rectangle) 鈥� (\begin{array}{l} S T^{2} + P S^{2} = P T^{2} \\ P S = Q R \end{array}) \\ = P T^{2} + (T R^{2} + Q R^{2}) 鈥勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌勨赌� (Rectangle) 鈥� (T R^{2} + Q R^{2} = Q T^{2}) \\ P Q^{2} = P T^{2} + Q T^{2} \end{matrix}$

Thus, as the square of the side $P Q$ is equal to the sum of squares of sides $P T$ and $Q T$ , this proves the Pythagoras theorem.

Therefore, the angle $P T Q$ is a right angle.

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