Explanation Formula: Area of segment: A s e g m e n t = A s e c t o r - A ∆ . Area of sector: If r is the radius of the circle, the area A of a sector whose arc has degree measure m is given by A = m 360 × π r 2 . 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: A = 1 2 b h . Sine ratio: In a right angled triangle, sine of an angle θ is given by: sin ⁡ θ = o p p ( θ ) h y p o t e n u s e Where o p p ( θ ) means the length of side opposite to θ . Calculation: Consider the following diagram in which the radius of inner circle is OC. Hence, O C = 3 in. The radius of outer circle is OA and OB. Hence, O A = O B = 6 in. We have to find the area of the segment formed by A B - and A B ^ . For finding the area of segment, first we need to find the area of sector formed by radii OA, OB and A B ^ To find the area of segment, first we have to find the degree measure of A B ^ which is equal to m ∠ A O B . Let’s consider right angled ∆ A O C . In this triangle, sin ⁡ ∠ O A C = O C O A = 3 6 = 1 2 We know that sin ⁡ 30 ° = 1 2 Hence, ∠ O A C = 30 ° Sum of all angles of a triangle is 180 ° In ∆ A O C , ∠ O A C + ∠ A C O + ∠ A O C = 180 ° After substituting the values of ∠ A C O and ∠ O A C , we get ∠ A O C = 60 ° Consider ∆ O C A and ∆ O C B . In these two triangles, O A = O B = 6 Hence, ∠ O C A = ∠ O C B = 90 ° O C is common side for both the triangles. From the above observations, it is clear that ∆ O C A ≅ ∆ O C B by RHS (right-angle hypotenuse side) test of congruency. If two triangles are congruent, then corresponding angles are equal. Hence, ∠ A O C = ∠ B O C Thus, ∠ B O C = 60 ° ∠ A O B = ∠ A O C + ∠ B O C = 60 + 60 = 120 ° The degree measure of A B ^ = m ∠ A O B = 120 °

Elementary Geometry for College Students

6th Edition

ISBN: 9781285195698

Author: Daniel C. Alexander, Geralyn M. Koeberlein

Publisher: Cengage Learning

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Polygon with three sides, three angles, and three vertices. Based on the properties of each side, the types of triangles are scalene (triangle with three three different lengths and three different angles), isosceles (angle with two equal sides and two equal angles), and equilateral (three equal sides and three angles of 60°). The types of angles are acute (less than 90°); obtuse (greater than 90°); and right (90°).

Chapter 8.5, Problem 26E

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The area of segment formed by chord of larger circle which is also a tangent of smaller circle.

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Chapter 8.5, Problem 25E

Chapter 8.5, Problem 27E

Chapter 8 Solutions

Elementary Geometry for College Students

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The area of segment formed by chord of larger circle which is also a tangent of smaller circle.

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To find: The area of segment formed by chord of larger circle which is also a tangent of smaller circle.

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Chapter 8 Solutions

To find:

The area of segment formed by chord of larger circle which is also a tangent of smaller circle.