Question

line segments mp and ml are perpendicular chords in circle o. mp = 10 and ml = 24.

which statements about circle o are true? select three options.

• the radius of circle o is 13.
• the diameter of circle o is 24.
• lp is a diameter of circle o.
• \angle lmp intercepts a semicircle.
• the measure of arc lp is 90^\circ.

Explanation:

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<reasoning>

Find the diameter of circle O

\[

$$\begin{aligned} &\angle LMP = 90^\circ \implies LP \text{ is a diameter of circle } O \\ &LP = \sqrt{MP^2 + ML^2} = \sqrt{10^2 + 24^2} = \sqrt{100 + 576} = \sqrt{676} = 26 \end{aligned}$$

\]

Find the radius of circle O

\[

$$\begin{aligned} &r = \frac{LP}{2} = \frac{26}{2} = 13 \end{aligned}$$

\]

Evaluate the statements

\[

$$\begin{aligned} &\text{1. The radius of circle } O \text{ is } 13. \quad (\text{True}) \\ &\text{2. The diameter of circle } O \text{ is } 24. \quad (\text{False, it is } 26) \\ &\text{3. } \overline{LP} \text{ is a diameter of circle } O. \quad (\text{True}) \\ &\text{4. } \angle LMP \text{ intercepts a semicircle.} \quad (\text{True, since } \angle LMP = 90^\circ) \\ &\text{5. The measure of arc } LP \text{ is } 90^\circ. \quad (\text{False, it is } 180^\circ) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>The radius of circle O is 13.</mcq-correct>
<mcq-option>The diameter of circle O is 24.</mcq-option>
<mcq-correct>LP is a diameter of circle O.</mcq-correct>
<mcq-correct>∠LMP intercepts a semicircle.</mcq-correct>
<mcq-option>The measure of arc LP is 90°.</mcq-option>
</answer>

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Answer:

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<reasoning>

Find the diameter of circle O

\[

$$\begin{aligned} &\angle LMP = 90^\circ \implies LP \text{ is a diameter of circle } O \\ &LP = \sqrt{MP^2 + ML^2} = \sqrt{10^2 + 24^2} = \sqrt{100 + 576} = \sqrt{676} = 26 \end{aligned}$$

\]

Find the radius of circle O

\[

$$\begin{aligned} &r = \frac{LP}{2} = \frac{26}{2} = 13 \end{aligned}$$

\]

Evaluate the statements

\[

$$\begin{aligned} &\text{1. The radius of circle } O \text{ is } 13. \quad (\text{True}) \\ &\text{2. The diameter of circle } O \text{ is } 24. \quad (\text{False, it is } 26) \\ &\text{3. } \overline{LP} \text{ is a diameter of circle } O. \quad (\text{True}) \\ &\text{4. } \angle LMP \text{ intercepts a semicircle.} \quad (\text{True, since } \angle LMP = 90^\circ) \\ &\text{5. The measure of arc } LP \text{ is } 90^\circ. \quad (\text{False, it is } 180^\circ) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>The radius of circle O is 13.</mcq-correct>
<mcq-option>The diameter of circle O is 24.</mcq-option>
<mcq-correct>LP is a diameter of circle O.</mcq-correct>
<mcq-correct>∠LMP intercepts a semicircle.</mcq-correct>
<mcq-option>The measure of arc LP is 90°.</mcq-option>
</answer>

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