Question

the 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 34. lp is a diameter of circle o. ∠lmp intercepts a semicircle. the measure of arc lp is 90°.

Explanation:

Step1: Use the Pythagorean theorem

Since \(MP = 10\) and \(ML=24\) and \(\angle PML = 90^{\circ}\), by the Pythagorean theorem \(LP^{2}=MP^{2}+ML^{2}\). Substitute \(MP = 10\) and \(ML = 24\) into the formula: \(LP^{2}=10^{2}+24^{2}=100 + 576=676\). Then \(LP=\sqrt{676} = 26\).

Step2: Analyze the radius

If \(LP\) is a diameter (by the inscribed - angle theorem: an inscribed angle of \(90^{\circ}\) intercepts a semicircle, so \(\angle PML = 90^{\circ}\) implies \(LP\) is a diameter). The radius \(r=\frac{LP}{2}\). Since \(LP = 26\), \(r = 13\).

Step3: Analyze the inscribed - angle and arc relationship

Since \(\angle PML=90^{\circ}\) and \(LP\) is a diameter (a diameter forms a semicircle), \(\angle LMP\) intercepts a semicircle.

Answer:

The radius of circle \(O\) is \(13\). \(\overline{LP}\) is a diameter of circle \(O\). \(\angle LMP\) intercepts a semicircle.