   Chapter 6.CR, Problem 32CR Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Solutions

Chapter
Section Elementary Geometry For College St...

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

A 30 ° - 60 ° - 90 ° triangle is inscribed in a circle with a radius of length 5 cm. Find the perimeter of the triangle.

To determine

To Find: The perimeter of the 30°-60°-90° triangle if it is inscribed in a circle with radius of length 5 cm.

Explanation

Definition:

A central angle of a circle is an angle whose vertex is the center of the circle and whose sides are radii of the circle.

An inscribed angle of a circle is an angle whose vertex is a point on the circle and whose sides are chords of the circle.

Theorems:

The measure of an inscribed angle of a circle is one-half the measure of its intercepted arc.

Congruent chords are located at the same distance from the center of a circle.

Angle inscribed in a semicircle is a right angle.

Calculation:

Given that a 30°-60°-90° triangle is inscribed in O with radius of length 5 cm.

Bytheorem,” Angle inscribed in a semicircle is a right angle”.

If a 30°-60°-90° triangle is inscribed in a circle, then its hypotenuse is diameter of the circle.

Let A,B, and C be vertices of the triangle.

And ABC=90°

Then AC is the diameters of the circle.

So, radiii = AO=CO=5 cm.

AB,BC, and CA are the sides of the triangle.

Since AB^:BC^=AB¯:BC¯

Let BAC=30°, AB^=60° and BCA=60°, BC^=120°

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