Explanation Given: Circle P has a radius of length 8 i n . Points A , B , C , and D lie on circle P in such a way that m ∠ A P B = 90 ° and m ∠ C P D = 60 ° . Theorem Used: According to the Pythagorean theorem, in a right-angled triangle, hypotenuse 2 = base 2 + perpendicular 2 . Property Used: If two sides of a triangle are equal then the angles formed by the legs of the included angle are also equal. The sum of all the angles of a triangle is 180 ° . An equilateral triangle has all the angles and all the sides equal. The perpendicular bisector divides the chord into two equal parts. Approach: i) Find the length of the chords, A B ¯ and C D ¯ . ii) Use the Pythagorean theorem to calculate the distance of chords from center (i.e.) P M ¯ and P N ¯ . iii) Find the difference between, P M ¯ and P N ¯ . Calculation: The central angles equals m ∠ A P B = 90 ° , m ∠ C P D = 60 ° and the radii equals P B ¯ = P C ¯ = 8 . In the given figure, Let M and N be perpendicular bisector on chords, A B ¯ and C D ¯ respectively. Applying Pythagorean theorem, in △ A P B since it is right angled, hypotenuse 2 =base 2 +perpendicular 2 . A B ¯ 2 = A P ¯ 2 + P B ¯ 2 or A B ¯ = A P ¯ 2 + P B ¯ 2 Substituting P B ¯ = A P ¯ = 8 in A B ¯ = A P ¯ 2 + P B ¯ 2 , A B ¯ = ( 8 ) 2 + ( 8 ) 2 = 64 + 64 = 128 = 8 2 In △ C P D , the vertex angle is m ∠ C P D = 60 ° , and sides P C ¯ = D P ¯ = 8 are equal so the base angles are also equal. Since, the sum of all the angles of a triangle is 180 ° . Hence, the sum of base angles is m ∠ P D C + m ∠ P C D = 180 ° − 60 ° = 120 ° Since, m ∠ P D C = m ∠ P C D , m ∠ P D C = m ∠ P C D = 120 ° 2 = 60 ° Thus, △ C P D is an equilateral triangle since all the angles are equal

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 6.4, Problem 31E

To determine

To find:

The distance that AB¯ is closer than CD¯ to point P.

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Chapter 6.4, Problem 30E

Chapter 6.4, Problem 32E

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