Chapter 8.1, Problem 43WE

To determine

To prove: Theangle PTQ is a right angle.

Explanation of Solution

Given:

Therectangle is shown below:

  

Side PS is the geometric mean between ST and TR.

  $P{S}^2=ST\times TR$

Calculation:

Consider side PQ,

  $PQ=SR\left(\text{Opposite}\text{side}\text{of}\text{rectangle}\right)$

Take square both sides as follows:

  $\begin{array}{c}P{Q}^2=S{R}^2\\ ={\left(ST+TR\right)}^2\\ =S{T}^2+ST\times TR+ST\times TR+T{R}^2\\ =\left(S{T}^2+P{S}^2\right)+\left(T{R}^2+P{S}^2\right)\left(\text{Rectangle}\right)\left(\begin{array}{l}S{T}^2+P{S}^2=P{T}^2\\ PS=QR\end{array}\right)\\ =P{T}^2+\left(T{R}^2+Q{R}^2\right)\left(\text{Rectangle}\right)\left(T{R}^2+Q{R}^2=Q{T}^2\right)\\ P{Q}^2=P{T}^2+Q{T}^2\end{array}$

Thus, as the square of the side PQ is equal to the sum of squares of sides PT and QT, this proves the Pythagoras theorem. Hence, the angle PTQ is a right angle.