Question
= 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
Given that triangle LMP is inscribed in circle O with \( MP = 10 \) and \( ML = 24 \), and that \( \angle PML \) is a right angle, we can use the properties of triangles inscribed in a circle to analyze the statements about circle O.
1. **Finding the length of LP:**
Since triangle LMP contains a right angle at M and is inscribed in circle O, we can use the Pythagorean theorem to find the length of side LP (the hypotenuse):
\[
LP^2 = ML^2 + MP^2
\]
\[
LP^2 = 24^2 + 10^2 = 576 + 100 = 676
\]
\[
LP = \sqrt{676} = 26
\]
2. **Finding the radius of circle O:**
The hypotenuse, LP, is the diameter of the circle because the angle \( \angle PML \) is a right angle. Thus, the radius \( r \) of circle O is:
\[
r = \frac{LP}{2} = \frac{26}{2} = 13
\]
3. **Finding the diameter of circle O:**
The diameter of circle O is simply the length of LP:
\[
\text{Diameter} = LP = 26
\]
Now let's evaluate the statements provided:
1. **The radius of circle O is 13.**
- This statement is **true**.
2. **The diameter of circle O is 34.**
- This statement is **false**. The diameter is 26, not 34.
3. **LP is a diameter of circle O.**
- This statement is **true**. As established, LP, being the hypotenuse of triangle LMP with a right angle at M, is the diameter.
4. **∠LMP intercepts a semicircle.**
- This statement is **true**. Since LP is the diameter, angle LMP intercepts the semicircle formed by points L and P on the circle.
5. **The measure of arc LP is 90°.**
- This statement is **false**. The measure of arc LP corresponds to the angular space created by point M at the ends of the diameter, which is actually 180°, not 90°.
Based on this analysis, the three true statements are:
1. The radius of circle O is 13.
2. LP is a diameter of circle O.
3. ∠LMP intercepts a semicircle.
1. **Finding the length of LP:**
Since triangle LMP contains a right angle at M and is inscribed in circle O, we can use the Pythagorean theorem to find the length of side LP (the hypotenuse):
\[
LP^2 = ML^2 + MP^2
\]
\[
LP^2 = 24^2 + 10^2 = 576 + 100 = 676
\]
\[
LP = \sqrt{676} = 26
\]
2. **Finding the radius of circle O:**
The hypotenuse, LP, is the diameter of the circle because the angle \( \angle PML \) is a right angle. Thus, the radius \( r \) of circle O is:
\[
r = \frac{LP}{2} = \frac{26}{2} = 13
\]
3. **Finding the diameter of circle O:**
The diameter of circle O is simply the length of LP:
\[
\text{Diameter} = LP = 26
\]
Now let's evaluate the statements provided:
1. **The radius of circle O is 13.**
- This statement is **true**.
2. **The diameter of circle O is 34.**
- This statement is **false**. The diameter is 26, not 34.
3. **LP is a diameter of circle O.**
- This statement is **true**. As established, LP, being the hypotenuse of triangle LMP with a right angle at M, is the diameter.
4. **∠LMP intercepts a semicircle.**
- This statement is **true**. Since LP is the diameter, angle LMP intercepts the semicircle formed by points L and P on the circle.
5. **The measure of arc LP is 90°.**
- This statement is **false**. The measure of arc LP corresponds to the angular space created by point M at the ends of the diameter, which is actually 180°, not 90°.
Based on this analysis, the three true statements are:
1. The radius of circle O is 13.
2. LP is a diameter of circle O.
3. ∠LMP intercepts a semicircle.