PQRS is a rectangle, the perpendicular ST from S on PR divides angle S in the ratio 2 : 3. Find angle TPQ.

Given: PQRS is a rectangle, the perpendicular ST from S on PR divides angle S in the ratio 2 : 3.

To find: Here we have to find the value of the angle TPQ.

Solution:

Now,

∠PSR = 90°

It is given that, the perpendicular ST from S on PR divides angle S in the ratio 2 : 3.

Divide the 90° angle into the ratio 2 : 3.

$2x\ +\ 3x\ =\ 90°$

$5x\ =\ 90°$

$x\ =\ 18°$

Therefore,

$2x\ =\ 2\ \times\ 18\ =\ 36°$

And,

$3x\ =\ 3\ \times\ 18\ =\ 54°$

So,

∠PST = 36°

Now, in ∆PST;

∠PST $+$ ∠PTS $+$ ∠TPS = 180°   (Sum of all angles of a triangle)

36° $+$ 90° $+$ ∠TPS = 180°  

∠TPS = 180° $-$ 36° $-$ 90° 

∠TPS = 180° $-$ 36° $-$ 90° 

∠TPS = 54°

Now, 

∠TPQ = ∠SPQ $–$ ∠TPS

∠TPQ = 90° $–$ 54° 

∠TPQ = 36°

So, value of angle TPQ is 36°.

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Updated on: 10-Oct-2022

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