<font face=TW Cen MT>Suppose a chord of a circle is 9 inches long and its midpoint is 6 inches from the center of the circle. Find the length of the radius.
How in the world do you figure that out?</font face=TW Cen MT>
You have a right triangle (centercircle, centercord, radius) You know two of the distances, solve for radius using the Pyh theorum .
To solve this problem, we can use the Pythagorean theorem.
Let's label the center of the circle as point O, the midpoint of the chord as point M, and the radius as r.
We are given that the chord is 9 inches long and its midpoint is 6 inches from the center of the circle. This means that OM, the distance from the center of the circle to the midpoint of the chord, is 6 inches.
Since OM is a perpendicular bisector of the chord, it divides the chord into two equal halves, each with a length of 4.5 inches.
Now, we can draw lines from O to the endpoints of the chord, creating two congruent right triangles, OMA and OMB. The hypotenuse of these triangles is the radius r, and the legs are OM and AM (or OM and BM).
Applying the Pythagorean theorem, we have:
r^2 = OM^2 + AM^2
We know that OM is 6 inches, and since the length of the chord is 9 inches, AM (or BM) is half of it, which is 4.5 inches.
Substituting these values into the equation, we get:
r^2 = 6^2 + 4.5^2
r^2 = 36 + 20.25
r^2 = 56.25
To find the length of the radius, we need to take the square root of both sides:
r = √56.25
r ≈ 7.5 inches
Therefore, the length of the radius is approximately 7.5 inches.