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two tangents tp and tq are drawn from an external point t to a circle they are inclined to each other at an angle of 100 degrees what is the measure of angle pqo?
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two tangents tp and tq are drawn from an external point t to a circle ...
Problem:
Two tangents TP and TQ are drawn from an external point T to a circle. They are inclined to each other at an angle of 100 degrees. What is the measure of angle PQO?
Solution:
Let O be the center of the circle.
Join OP and OQ.
Now, TP and TQ are tangents to the circle.
Therefore, OP and OQ are perpendicular to TP and TQ respectively.
So, angle PTO = 90 degrees and angle QTO = 90 degrees.
Also, angle TPO = angle TQO = 90 degrees (since TP and TQ are tangents).
Therefore, triangles TPO and TQO are right-angled triangles.
Thus, by trigonometry, we have:
tan (angle OPT) = TP/OP and tan (angle OQT) = TQ/OQ
Now, angle PTO = 90 degrees and angle QTO = 90 degrees.
Therefore, angle OPT + angle OQT = 180 - angle PQO (sum of angles in triangle OPQ).
Thus, we have:
tan (angle OPT) + tan (angle OQT) = TP/OP + TQ/OQ
Now, we are given that angle TPQ = 100 degrees.
Therefore, angle OPT + angle OQT = 180 - angle TPQ = 80 degrees.
Also, we know that OP = OQ (since O is the center of the circle).
Thus, we have:
tan (angle OPT) + tan (angle OQT) = 2TP/OP (since TQ = TP)
Putting the values of tan (angle OPT) + tan (angle OQT) and simplifying, we get:
tan (40 degrees) = TP/OP
Thus, angle PTO = 90 degrees and angle OPT = 40 degrees.
Therefore, angle OQP = angle OPT - angle TPQ = 40 - 100 = -60 degrees.
But angle OQP cannot be negative.
Therefore, we take angle OQP = 360 - 60 = 300 degrees.
Thus, angle PQO = angle PQT + angle TQO = 100 + 90 = 190 degrees.
Answer:
The measure of angle PQO is 190 degrees.
Community Answer
two tangents tp and tq are drawn from an external point t to a circle ...
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