For concentric circles with radii of lengths in. and in., find the exact area of the smaller segment determined by a chord of the larger circle that is also a tangent of the smaller circle.
The area of segment formed by chord of larger circle which is also a tangent of smaller circle.
Area of segment:
Area of sector:
If r is the radius of the circle, the area of a sector whose arc has degree measure m is given by
Area of triangle:
If b and h are lengths of base and height of triangle respectively, then area A of triangle is given by the formula:
In a right angled triangle, sine of an angle is given by:
Where means the length of side opposite to .
Consider the following diagram in which the radius of inner circle is OC. Hence, in.
The radius of outer circle is OA and OB. Hence, in.
We have to find the area of the segment formed by and .
For finding the area of segment, first we need to find the area of sector formed by radii OA, OB and
To find the area of segment, first we have to find the degree measure of which is equal to .
Let’s consider right angled . In this triangle,
We know that
Sum of all angles of a triangle is
After substituting the values of and , we get
Consider and .
In these two triangles,
is common side for both the triangles.
From the above observations, it is clear that by RHS (right-angle hypotenuse side) test of congruency.
If two triangles are congruent, then corresponding angles are equal.
The degree measure of
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