Question

QTR is a right triangle TQR is a right triangle QS is an altitude which statements are true

Answer 1

4)

`(TQ)^2+(QR)^2=(ST)(TR)+(SR)(TR)`

Explanation:

`m\angle QTR=m\angle SQR\\</p><p>m\angle TSQ=m\angle TQR=90°\\</p><p>\implies \Delta TSQ \sim \Delta TQR\\</p><p>\implies (TQ)/(TR)=(TS)/(TQ)\\</p><p>\implies \boxed{(TQ)^2= (TS)(TR)}\\</p><p>\text{Similarly, use triangles RSQ and RQT to get:}\\</p><p>\boxed{(QR)^2=(SR)(TR)}`

Now add the two equations to get the answer.

Answer 2

(TQ)² + (QR)² = (ST) (QS) + (SR) (QS)

Explanation:

PROOF:

QTR is a right triangle TQR is a right triangle QS is an altitude

From right angle theorem

Side √TQ² = √(QS² + ST²) reason ∠QST is right angled

Side √QR² = √(SR² + QS²) reason ∠RSQ is right angled

QED