Raidus of a Circle With Given Information

Question

A chord is 48cm and another chord is 40cm. The 48cm chord is 8cm closer to the center of the circle than the 40cm chord. What is the radius of the circle?

Answer 1

Call the radius "r".

Draw the 48 cm chord. Draw radii to both endpoints of the chord, and a third radius that is the perpendicular bisector to the chord. This forms two congruent triangles. They have sides r (the hypotenuse), x (the distance from the center of the circle to the chord, and 24.

Now do the same thing for the other chord. It will have sides r (the hypotenuse), x+8 (the distance from the center of the circle to the chord, and 20.

Now we use the Pythagorean theorem twice:

r^2 = (x+8)^2 + 20^2

r^2 = x^2 + 24^2

Since thee two equations are both equal to r^2, we can just write one equation with "x" in it:

(x+8)^2 + 400 = x^2 + 576

now expand the left hand side (x+8)^2 term using FOIL or whatever you like:

(x^2 + 16x + 64) + 400 = x^2 + 576

subtract x^2 and 464 from both sides:

16x = 112

Divide each side by 7:

x = 7

Your two triangles are a 7-24-r triangle and a 15-20-r triangle, where r is the hypotenuse. Pick your favorite one and you'll see that r, the radius of the circle that we needed to find, is 25 cm. (15-20-25 is just a 3-4-5 triangle stretched by a factor of 5. 7-24-25 is a pythagorean triple all by itself).