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: $∠ P T Q$ is a right angle.

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information.

The rectangle is shown below,

Step 2. Concept used.

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

Then,

$P S^{2} = S T \times T R$

Consider side $P Q$ ,

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

Step 3. Take square both sides.

$\begin{matrix} P Q^{2} = S R^{2} \\ = {(S T + T R)}^{2} \\ = S T^{2} + S T \times T R + S T \times T R + T R^{2} \\ = (S T^{2} + P S^{2}) + (T R^{2} + P S^{2}) (Rectangle) (\begin{array}{l} S T^{2} + P S^{2} = P T^{2} \\ P S = Q R \end{array}) \\ = P T^{2} + (T R^{2} + Q R^{2}) (Rectangle) (T R^{2} + Q R^{2} = Q T^{2}) \\ P Q^{2} = P T^{2} + Q T^{2} \end{matrix}$

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Concept used.

Step 3. Take square both sides.

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Given: PQRSis a rectangle.PS is the geometric mean between ST and TR.Prove: ∠PTQ is a right angle.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Concept used.

Step 3. Take square both sides.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Statistics

Mechanics Maths

Probability and Statistics

Geometry

Calculus

Study anywhere. Anytime. Across all devices.

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