
In the figure, O is the center of the circle and . If , find and .

Answer
494.1k+ views
Hint: First we will join BC in the given diagram. Then we will consider triangle and use that to prove it is an isosceles triangle and the two angles inclined on equal and non-equal lines are same. Then we will use the angle sum property of triangle in the triangle ABC, where the sum of the interior angles is and then use the central angle theorem to find the required values.
Complete step by step answer:
We are given that the O is the center of the circle, and .
Joining BC in the given diagram, we get
First, we will consider triangle .
We are given that , it means that triangle ABC is an isosceles triangle.
We know that in an isosceles triangle, the two angles inclined on equal and non-equal lines are the same.
Then we will use the angle sum property of triangle in the triangle ABC, where the sum of the interior angles is , we get
Substituting the value of and in the above equation, we get
Subtracting the above equation by on both sides, we get
Now according to the central angle theorem, the central angle from any two points on the circle is always twice the inscribed angles from those two points, so we have
Using the central angle theorem again, we get
Dividing the above equation by 2 on both sides, we get
Therefore, the measure of is and is .
Note: In solving these types of questions, you should be familiar with the concept of angle of elevation and the tangential properties. Students should not use any other trigonometric functions like sine or cosine of the given angle because the given condition of the problem, which is provided to us, is of the height of the triangle and a base. So, do not take the sum equal to instead , which is wrong.
Complete step by step answer:
We are given that the O is the center of the circle,
Joining BC in the given diagram, we get

First, we will consider triangle
We are given that
We know that in an isosceles triangle, the two angles inclined on equal and non-equal lines are the same.
Then we will use the angle sum property of triangle in the triangle ABC, where the sum of the interior angles is
Substituting the value of
Subtracting the above equation by
Now according to the central angle theorem, the central angle from any two points on the circle is always twice the inscribed angles from those two points, so we have
Using the central angle theorem again, we get
Dividing the above equation by 2 on both sides, we get
Therefore, the measure of
Note: In solving these types of questions, you should be familiar with the concept of angle of elevation and the tangential properties. Students should not use any other trigonometric functions like sine or cosine of the given angle because the given condition of the problem, which is provided to us, is of the height of the triangle and a base. So, do not take the sum equal to
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