   Chapter 6.CR, Problem 33CR 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 circle is inscribed in the right triangle. The length of the radius of the circle is 6 cm, and the length of the hypotenuse is 29 cm. Find the lengths of the two segments of the hypotenuse that are determined by the point of tangency.

To determine

To Find: The lengths of two segments of a 29 cm long hypotenuse of the triangle that are determined by the point of tangency, if a circle with radius length 6 cm is inscribed in the right triangle.

Explanation

Concept:

Theorem 6.3.4:

The tangent segments to a circle from an external point are congruent.

Calculation:

Given that a circle O is inscribed in a right triangle.

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

And ABC=90°

Then AC = 29 cm is the hypotenuse of the triangle.

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

Also, AB,BC,  and CA are the tangents of the circle at R, Q and P respectively.

By Theorem 6.3.4:

The tangent segments to a circle from an external point are congruent.

ROBQ is a square.

So, radiii =RO=QO=BO=OP=6 cm.

And BQ=BR=6 cm

So, let AR-=AP-=x and PC-=QC-=y

Because ABC is right triangle, AC2=AB2+BC2

(x+y)2=(x+6)2+(y+6)2

x2+2xy+y2=x2+12x+36+y2+12y+36

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