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Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

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BuyFindarrow_forward

Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

In P , whose radius has length 8 i n . , m A B = m B C = 60 ° . Because m A C = 120 ° chord A C ¯ is longer than either of the congruent chords A B ¯ and B C ¯ . Determine how much longer A C ¯ is than A B ¯ ; that is, find the exact value and the approximate value of A C A B .

To determine

To find:

The exact value and the approximate value of ACAB.

Explanation

Given:

P, whose radius has length 8 in., mAB=mBC=60°. mAC=120° and chord AC¯ is longer than either of the congruent chords AB¯ and BC¯.

The figure given below,

Property Used:

The measure of an inscribed angle of a circle is half the measure of its intercepted arc. Consider the figure below, let the inscribed angle be 2.

Then m2=12mHJ and generally HKKJ.

According to the Pythagorean theorem, in a right-angled triangle

hypotenuse2=base2+perpendicular2.

Calculation:

The measure of an inscribed angle of a circle is half the measure of its intercepted arc and from the given figure,

mCAB=12mBC=12(60°)=30°

mBCA=12mAB=12(60°)=30° and

mCBA=12(360°mAC)=12(360°120°)=12(240°)=120°.

Since, two angles are equal in the triangle, hence it is an isosceles triangle. So, the two sides corresponding legs will be equal i.e. AB¯=BC¯.

Let L, M, and N be perpendicular bisector on chords AB¯, BC¯ and AC¯ respectively as a radius perpendicular to a chord bisects the chord and its arc.

Since, radii are equal AP=PB we get the angles also equal i.e.

PAB=PBA=180°60°2=60°

Also, APB=mAB=60°. Since all angles are equal, APB is equilateral triangle.

Thus, mPB¯=mPA¯=mAB¯=8.

Similarly, radii are equal PB=PC we get the angles also equal i.e.

PBC=PCB=180°60°2=60°

Also, BPC=mBC=60°

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