Question

Triangle PQR right angled at R, M is the midpoint of PR, then prove that PR^2=4(QM^2-QR^2)

Answer 1

Answer:

Secondary School Math 13+7 pts

In a triangle PQR, PR2-PQ2 =QR2 and M is a point on side PR such that QM is perpendicular to PR. Prove that QM2=PM*MR

Report by Mitali8368 12.02.2018

Answers

nikitasingh79

Nikitasingh79★ Brainly Teacher ★

[FIGURE IS IN THE ATTACHMENT]

Given:

In ∆ PQR, PR²-PQ²= QR² & QM ⊥ PR

To Prove: QM² = PM × MR

Proof:

Since, PR² - PQ²= QR²

PR² = PQ² + QR²

So, ∆ PQR is a right angled triangle at Q.

In ∆ QMR & ∆PMQ

∠QMR = ∠PMQ [ Each 90°]

∠MQR = ∠QPM [each equal to (90°- ∠R)]

∆ QMR ~ ∆PMQ [ by AA similarity criterion]

By property of area of similar triangles,

ar(∆ QMR ) / ar(∆PMQ)= QM²/PM²

1/2× MR × QM / ½ × PM ×QM = QM²/PM²

[ Area of triangle= ½ base × height]

MR / PM = QM²/PM²

QM² × PM = PM² × MR

QM² =( PM² × MR)/ PM

QM² = PM × MR