To determine the radius of Circle P given a chord of length 8 inches and the perpendicular distance from the center to the chord being 3 inches, follow these steps:
1. Understand the Geometry:
- Draw circle P with the center labeled as O.
- Draw a chord AB of length 8 inches.
- Draw a perpendicular line from the center O to the chord AB, meeting AB at point M.
The length OM is the distance from the center to the chord, which is given as 3 inches.
2. Divide the Chord:
- Since OM is the perpendicular from the center to the chord AB, it bisects AB. Hence, AM and MB are each half the length of AB.
- Thus, AM = MB = [tex]\( \frac{8}{2} = 4 \)[/tex] inches.
3. Form a Right Triangle:
- Consider the right triangle OMA where OM is one leg (3 inches), MA is the other leg (4 inches), and OA (the radius) is the hypotenuse.
4. Apply Pythagoras' Theorem:
- According to the Pythagorean theorem, in the right triangle OMA:
[tex]\[
OA^2 = OM^2 + MA^2
\][/tex]
- Substituting the given lengths:
[tex]\[
OA^2 = 3^2 + 4^2
\][/tex]
- Calculate the squares:
[tex]\[
OA^2 = 9 + 16
\][/tex]
- Add the results:
[tex]\[
OA^2 = 25
\][/tex]
5. Solve for OA (the Radius):
- Take the square root of both sides to find OA:
[tex]\[
OA = \sqrt{25} = 5
\][/tex]
Thus, the radius of Circle P is 5 inches.
