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Maths-

General

Easy

Question

In $straight triangle P Q R comma Q M perpendicular R P$ and $P R squared minus P Q squared equals Q R squared$ prove that $QM squared equals PM cross times MR.$

The correct answer is: Hence proved.

Solution :-
Aim :- Prove that $Q M squared equals P M cross times M R$

Hint :- using inverse of pythagoras theorem, we get right angle at Q,
Now using angles prove the similarity of QMR and PMQ triangles. And find the
ratios to get the desired property
Explanation(proof ) :-
Given , $P R squared minus P Q squared equals Q R squared not stretchy rightwards double arrow P R squared equals P Q squared plus Q R squared$
Here ,PR is hypotenuse and angle Q is right angle.
Then $straight angle P equals 90 minus straight angle R$
$straight angle P equals 90 minus straight angle R text So we get end text straight angle P equals straight angle M Q R$
We know $straight angle M equals straight angle M equals 90 left parenthesis text degrees end text right parenthesis$
By AA similarity

$straight capital delta Q M R tilde straight triangle P M Q$

By similarity we get $fraction numerator Q M over denominator P M end fraction equals fraction numerator R M over denominator M Q end fraction not stretchy rightwards double arrow Q M squared equals P M cross times M R$
Hence proved.

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