Question

a circle having a centre O has a triangle PQS inside it with the base QP is not on centre O. PT and QT are two tangents from an exterior point T (S, O and T are linear points - lie in one line )
it is given that angle ptq = 25 deg . find angle PSQ and OPQ

Answer 1

Is this how the diagram looks? See attachment.

Answer 2

yes

Answer 3

Angle QTP is an angle formed by two tangents to a circle from an outside point.

The formula for calculating its measure is shown on the attachment.

Answer 4

In the case of two tangents to a circle, the near arc and the far arc together form the entire circle.

Answer 5

The near arc is red; the far arc is blue in the diagram for the problem.

Answer 6

If the red arc has measure R, then the blue arc has measure 360 - R.

25 = 1/2[ (360 -R) - R ]

Solve for R.

@msingh

Answer 7

50=360-2R
2R=360-50
2R=310
R=155

Answer 8

Correct. Notice that the near red arc and the angle formed by the tangents sums to 180. That will always be the case in this type scenario.

Answer 9

Look at angle S. It is an inscribed angle of a circle and has measure half of its intercepted arc. Angle S intercepts the red arc.

What is the measure of angle S ? @msingh

Answer 10

155

Answer 11

Try again.

Hint: inscribed angle of a circle and has measure **half** of its intercepted arc

Answer 12

@msingh

Answer 13

oopssrry
155/2

Answer 14

Tangents drawn to a circle from an outside point are congruent.
That makes triangle PQT isosceles with segment TQ congruent to segment TP.

Answer 15

Looking at triangle PQT, recall that if two sides of a triangle are congruent, then the angles opposite those sides are congruent.

Answer 16

Angle TQP has the same measure as Angle TPQ.
The sum of the interior angles of a triangle is 180.
Angle T was given to be 25.
So, what is the degree measure of angle TPQ? @msingh

Answer 17

180-25=2x
155/2=x
where x= angle TOP =angleTPQ

Answer 18

Yes.

Answer 19

Note on the attached diagram that segment OP is a radius of the circle.

Also, recall that if a radius is drawn to a tangent to a circle at the point of tangency, the radius is perpendicular to the tangent.

Answer 20

Angle TPO is a right angle. Angle TPQ and Angle QPO form that right angle, making the sum of those measures 90. Angle TPQ is 77.5 or 155/2 so what is the measure of angle OPQ ? @msingh

Answer 21

90-77.5

Answer 22

I think I made a typo on the angle.
Regardless, you read my mind.

Angle OPQ is 22.5

I think the problem is solved.

Answer 23

k

Answer 24

Questions?

Answer 25

till now , no questions
but wanna say thank u 2 u @Directrix

Answer 26

Happy to help.