   Chapter 8.5, Problem 45E 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 triangle has sides of lengths 6 cm, 8 cm and 10 cm. Find the distance between the center of the inscribed circle and the center of the circumscribed circle for this triangle. Give the answer to the nearest tenth of a centimetre.

To determine

To find:

The distance between centre of inscribed circle and centre of circumscribed circle.

Explanation

Formula:

Radius of inscribed circle for right angled triangle:

If a, b and c are the lengths of sides of right angled triangle (c is the length of hypotenuse), then the length of radius r of circle inscribed in a right angled triangle is r=aba+b+c.

Circumscribed circle of triangle:

The unique circle that passes through each of the triangle's three vertices is the circumscribed circle of triangle. The centre of this circle is called the circumcentre.

The circumcentre is the point of intersection of perpendicular bisectors of all the three sides of triangle.

In special case of a right angled triangle, circumcentre lies exactly at the midpoint of hypotenuse.

Calculation:

Let’s check whether the triangle is a right angled triangle or not.

From the given lengths of sides of triangle, the longest side measures 10.

102=100.

The sum of squares of remaining two sides =82+62=64+36=100

Thus, 102=82+62.

Hence, the triangle is a right angled triangle with the length of hypotenuse =10.

Let, a=6, b=8 and c=10.

Let’s substitute these values in the formula to find the radius r of inscribed circle.

r=aba+b+c

r=6×86+8+10

r=4824

r=2

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