Question

line segments mp and ml are perpendicular chords in circle o. mp = 10 and ml = 24. which statements about circle o are true? choose three correct answers. the radius of circle o is 13. ∠lmp intercepts a semicircle. the diameter of circle o is 34. the measure of arc lp is 90°. lp is a diameter of circle o.

Explanation:

Step1: Apply Pythagorean theorem

Since \(MP\) and \(ML\) are perpendicular chords and \(LP\) is the diameter (angle in a semicircle is a right - angle, so \(\angle M = 90^{\circ}\) and \(LP\) is the hypotenuse of right - triangle \(MLP\)). Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 10\), \(b = 24\), and \(c=LP\). So \(LP=\sqrt{10^{2}+24^{2}}=\sqrt{100 + 576}=\sqrt{676}=26\).

Step2: Find the radius

The radius \(r=\frac{LP}{2}\). Since \(LP = 26\), then \(r = 13\).

Step3: Check the diameter property

Since \(\angle LMP=90^{\circ}\), by the inscribed - angle theorem (an inscribed angle that intercepts a semicircle is a right angle and vice - versa), \(\angle LMP\) intercepts a semicircle (the arc \(LP\)). And \(LP\) is a diameter of the circle.

Answer:

The radius of circle \(O\) is \(13\); \(\angle LMP\) intercepts a semicircle; \(LP\) is a diameter of circle \(O\)