Question
Line segments MP and ML are perpendicular chords in circle O. MP = 10 and ML = 24.
Triangle L M P is inscribed within circle O. The length of M P is 10 and the length of M L is 24. Angle P M L is a right angle.
Which statements about circle O are true? Select three options.
The radius of circle O is 13.
The diameter of circle O is 34.
LP is a diameter of circle O.
∠LMP intercepts a semicircle.
The measure of arc LP is 90°.
Triangle L M P is inscribed within circle O. The length of M P is 10 and the length of M L is 24. Angle P M L is a right angle.
Which statements about circle O are true? Select three options.
The radius of circle O is 13.
The diameter of circle O is 34.
LP is a diameter of circle O.
∠LMP intercepts a semicircle.
The measure of arc LP is 90°.
Answers
GPT-4o mini
To analyze the statements about circle O given the chords MP and ML, we can use the properties of circles and the Pythagorean theorem since we're dealing with a right triangle.
1. **Chords and Right Angles**: Since MP and ML are perpendicular, triangle LMP forms a right triangle with legs ML = 24 and MP = 10. We can calculate the hypotenuse (which will serve as the distance from O to point P and L) using the Pythagorean theorem:
\[
OL^2 = OM^2 + ML^2
\]
\[
OP^2 = OM^2 + MP^2
\]
We can also denote the lengths from the center O to the chord MP and ML as \( d_1 \) and \( d_2 \), respectively.
From our calculations:
\[
d_1 = OM = 24 / 2 = 12 \quad (\text{distance from center O to midpoint of chord ML})
\]
\[
d_2 = OM = 10 / 2 = 5 \quad (\text{distance from center O to midpoint of chord MP})
\]
In the right triangle LMP, the radius (r) can be found using:
\[
r^2 = OM^2 + \left(\frac{ML}{2}\right)^2 = OM^2 + 12^2
\]
\[
r^2 = 5^2 + 12^2 = 25 + 144 = 169
\]
Thus, the radius \( r = \sqrt{169} = 13 \).
2. **Diameter**: The diameter \( D \) is twice the radius:
\[
D = 2r = 2 \times 13 = 26.
\]
3. **Additional Properties**:
- Since LP spans from L to P along the circle and both endpoints are at the ends of the respective lengths of the segments, it doesn’t directly relate to diameter properties established through midpoint calculations.
- \(\angle LMP\) being a right angle indicates that it intercepts an arc that is a semicircle since any angle inscribed in a semicircle is a right angle.
- The measure of arc LP that corresponds to the right angle should be 90° as that corresponds to the 1/4 circle intercepted by the angle.
Given the results of our analysis, the true statements about circle O are:
1. The radius of circle O is 13.
2. \(\angle LMP\) intercepts a semicircle.
3. The measure of arc LP is 90°.
Thus, the true statements are:
- The radius of circle O is 13.
- ∠LMP intercepts a semicircle.
- The measure of arc LP is 90°.
(Note: The diameter of circle O mentioned in the second point should be adjusted since it was initially calculated incorrectly; it is not 34 and should be noted as 26, thus making that option incorrect.)
1. **Chords and Right Angles**: Since MP and ML are perpendicular, triangle LMP forms a right triangle with legs ML = 24 and MP = 10. We can calculate the hypotenuse (which will serve as the distance from O to point P and L) using the Pythagorean theorem:
\[
OL^2 = OM^2 + ML^2
\]
\[
OP^2 = OM^2 + MP^2
\]
We can also denote the lengths from the center O to the chord MP and ML as \( d_1 \) and \( d_2 \), respectively.
From our calculations:
\[
d_1 = OM = 24 / 2 = 12 \quad (\text{distance from center O to midpoint of chord ML})
\]
\[
d_2 = OM = 10 / 2 = 5 \quad (\text{distance from center O to midpoint of chord MP})
\]
In the right triangle LMP, the radius (r) can be found using:
\[
r^2 = OM^2 + \left(\frac{ML}{2}\right)^2 = OM^2 + 12^2
\]
\[
r^2 = 5^2 + 12^2 = 25 + 144 = 169
\]
Thus, the radius \( r = \sqrt{169} = 13 \).
2. **Diameter**: The diameter \( D \) is twice the radius:
\[
D = 2r = 2 \times 13 = 26.
\]
3. **Additional Properties**:
- Since LP spans from L to P along the circle and both endpoints are at the ends of the respective lengths of the segments, it doesn’t directly relate to diameter properties established through midpoint calculations.
- \(\angle LMP\) being a right angle indicates that it intercepts an arc that is a semicircle since any angle inscribed in a semicircle is a right angle.
- The measure of arc LP that corresponds to the right angle should be 90° as that corresponds to the 1/4 circle intercepted by the angle.
Given the results of our analysis, the true statements about circle O are:
1. The radius of circle O is 13.
2. \(\angle LMP\) intercepts a semicircle.
3. The measure of arc LP is 90°.
Thus, the true statements are:
- The radius of circle O is 13.
- ∠LMP intercepts a semicircle.
- The measure of arc LP is 90°.
(Note: The diameter of circle O mentioned in the second point should be adjusted since it was initially calculated incorrectly; it is not 34 and should be noted as 26, thus making that option incorrect.)