Question

In figure, PQR is a right triangle right angled at Q and QS ⊥ PR. If PQ = 6 cm and PS = 4 cm, then perimeter of ΔQSR is ______.

Answer 1

In figure, PQR is a right triangle right angled at Q and QS ⊥ PR. If PQ = 6 cm and PS = 4 cm, then perimeter of ΔQSR is `underlinebb(5(sqrt(5) + 1) cm)`.

Explanation:

In a right triangle with altitude QS to the hypotenuse PR we have the standard relations PQ² = PR·PS, QS² = PS·SR and QR² = PR·SR. 

Given PQ = 6 and PS = 4

PQ² = 36 = PR·4

So PR = 9. 

Hence SR = 9 – 4 = 5.

Then `QS = sqrt((PS·SR)` 

= `sqrt(4·5)`

= `2sqrt(5)` 

And `QR = sqrt(PR·SR)`

= `sqrt(9·5)`

= `3sqrt(5)` 

Perimeter (ΔQSR) = QS + SR + QR 

= `2sqrt(5) + 5 + 3sqrt(5)` 

= `5 + 5sqrt(5)` cm