Question

P Q R

is a triangle right angle at

P

and

M

is a point on

Q R

such that

P M ⊥ Q R

. Show that

P M^{2} = Q M \times M R

.

Accepted Answer

Given, $P Q R$ is a triangle right angle at $P$ and $M$ is a point on $Q R$ such that $P M ⊥ Q R$ .

To prove: $P M^{2} = Q M \times M R$

Proof: Pythagoras theorem states that in a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides.

In $△ P R M$ ,

$P R^{2} = P M^{2} + M R^{2}$ ___________ $(1)$

In $△ P M Q$ ,

$P Q^{2} = P M^{2} + M Q^{2}$ _____________ $(2)$

In $△ P Q R$ ,

$P Q^{2} + P R^{2} = Q R^{2}$

$\Rightarrow P Q^{2} + P R^{2} = {(R M + M Q)}^{2}$

$\Rightarrow P Q^{2} + P R^{2} = R M^{2} + M Q^{2} + 2 \times R M \times M Q$ ________________ $(3)$

Adding equation $(1)$ and $(2)$ , we get,

$\Rightarrow P Q^{2} + P R^{2} = R M^{2} + M Q^{2} + P M^{2} + P M^{2}$

$\Rightarrow P Q^{2} + P R^{2} = R M^{2} + M Q^{2} + 2 P M^{2}$ ________________ $(4)$

From equation $(3)$ and $(4)$ , we get,

$\Rightarrow R M^{2} + M Q^{2} + 2 P M^{2} = R M^{2} + M Q^{2} + 2 \times R M \times M Q$

$\Rightarrow 2 P M^{2} = 2 \times R M \times M Q$

$\Rightarrow P M^{2} = R M \times M Q$

Hence, proved.

Andhra Pradesh Board Solutions for Chapter: Similar Triangles, Exercise 14: EXERCISE - 8.4

Andhra Pradesh Board Mathematics Solutions for Exercise - Andhra Pradesh Board Solutions for Chapter: Similar Triangles, Exercise 14: EXERCISE - 8.4

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