Question

Given that RS=18 and QR =6, identify the length of PS.

Answer 1

There are 3 right-angled triangles in the diagram. PSR, PRQ, and SPQ. Thus, we can apply the Pythagoras theorem to solve the question.

`\begin{gathered} PQ^2=PR^2+QR^2\text{ (}\Delta\text{PQR)} \\ QR=6 \\ PQ^2=PR^2+6^2 \\ PQ^2=PR^2+36\text{ (Equation 1)} \\ \\ PQ^2+PS^2=24^2\text{ (}\Delta\text{SPQ)} \\ PQ^2+PS^2=576\text{ (Equation 2)} \\ \\ PS^2=PR^2+RS^2\text{ (}\Delta\text{PRS)} \\ PS^2=PR^2+18^2 \\ PS^2=PR^2+324\text{ Equation (3)} \end{gathered}`

Solving all three equations simultaneously, we have:

`\begin{gathered} Substitute\text{ the expression for PQ from Equation 1 into Equation 2} \\ \text{Equation 2:} \\ PQ^2+PS^2=576 \\ PR^2+36+PS^2=576 \\ \text{Subtract 36 from both sides} \\ PR^2+PS^2=576-36=540 \\ \\ PR^2+PS^2=540\text{ (Equation 4)} \\ Make\text{ PR}^2\text{ the subject of the formula} \\ PR^2=540-PS^2 \\ \\ \text{Substitute the expression for PR}^2\text{ to Equation }3 \\ PS^2=540-PS^2_{}+324 \\ \text{Add PS}^2\text{ to both sides} \\ PS^2+PS^2=864 \\ 2PS^2=864 \\ \text{Divide both sides by 2} \\ PS^2=432 \\ \text{Square root both sides} \\ PS=\sqrt[]{432}=12\sqrt[]{3} \end{gathered}`

Final Answer

The answer is:

`PS=12\sqrt[]{3}`