Question

Given: $PQRS$is a rectangle.

$PS$ is the geometric mean between $ST$ and $TR$.

Prove: $\angle PTQ$ is a right angle.

Short answer

The angle $PTQ$ is a right angle.

Step-by-step solution

Step 1. Given information.

The rectangle is shown below,

Step 2. Concept used.

Side $PS$is the geometric mean between $ST$and $TR$.

Then,

$P{S}^2=ST\times TR$

Consider side $PQ$,

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

Step 3. Take square both sides.

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

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.

Therefore, the angle $PTQ$ is a right angle.