
Let triangle ABC have vertices on a circle. Let AD be the altitude and AP be the diameter of the circle. If and then equals to
A.
B.
C.
D.

Answer
494.7k+ views
Hint: We join BP and CP. We use the theorem that the sum of the angles in a triangle is in triangles ABD, ACD to get . We use the theorem that an arc of semi-circular length always subtends right angle of measure to get . We use the theorem that angle subtended by the same arc have equal measures to get and then we find the required angle .
Complete step-by-step solution
We are given in the figure a triangle ABC whose vertices are on a circle. AD is the altitude dropped on the side BC and AP is the diameter of the circle. We are further given the measure of angles and . We are asked to find a measure of . Let us join BP and CP and have the figure below.
Since AD is the altitude we have . We know that sum of the angles in a triangle is . So we have in triangle ADB
We also have in triangle ADC,
We are given that AP is a diameter which means AP divides the circle into two semi-circles. We know that an arc of semi-circular length always subtends the right angle of measure at any point on the circle. Here the semicircle has subtended and on the points on the circle respectively. So we have,
Then we have
We know that angles subtended by the same arc have equal measures. Here in the circle arc BP subtends angles . So we have,
Then we have
So the correct option is D.
Note: We can alternatively solve using the sum of the angles in triangle is to get in triangle . We then join OC and use the theorem that the central angle of an arc is twice the angle subtended on appointing on the circle but not on the arc for the arc AC to get and then the required angle .
Complete step-by-step solution
We are given in the figure a triangle ABC whose vertices are on a circle. AD is the altitude dropped on the side BC and AP is the diameter of the circle. We are further given the measure of angles

Since AD is the altitude we have
We also have in triangle ADC,
We are given that AP is a diameter which means AP divides the circle into two semi-circles. We know that an arc of semi-circular length always subtends the right angle of measure
Then we have
We know that angles subtended by the same arc have equal measures. Here in the circle arc BP subtends angles
Then we have
So the correct option is D.
Note: We can alternatively solve using the sum of the angles in triangle is
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