In searching an ancient ruin an archeologist found a piece o

In searching an ancient ruin an archeologist found a piece of the rim of Ban old wheel. In order to restore the wheel he needed the diameter. He marked three points, A, B, C, on the rim so that AB was congruent to AC. B = 15 and BC = 24, what was the diameter of the wheel?

Solution

The given conditions are

1. AB is congruent to AC, it means it is an isoceles triangle

2. AB = AC = 15 & BC =24

so based on the given condition we can assume the figure would be like the following

R = Center of the rim. D is the Midpoint of AC and AD is perpendicular to BC as it is an isoceles triangle

using pythagorus theorem for triangle ABD, we get

AB² = AD² + BD²

15² = AD² + 12²

225-144 = AD²

AD = 9

Let radius of rim be denoted by r, then we have

AD = AR (Radius) + RD = r + RD

RD = 9 - r   ........................................................................(1)

Now using pythagorus theorem for triangle BDR, we get

BR² = RD² + BD²

BR = r (Radius), RD = 9 - r, BD = 12, on solving we get

r2 = (9 - r)2 + 144

r2 = 81 - 18r + r2 + 144

18r = 225

r = 12.5

Radius of the rim is 12.5 and diameter is 25.

So the arrived figure would be as follows