Question

the measure of central angle lop is °. the length of lp is units. the length of mp is units. (options for angle: 45, 90, 180, 360; diagram: circle with center o, points l, m, p on circle, lm=8, right angle at m, op=8.5)

Explanation:

Step1: Determine central angle LOP

LP is a diameter (since O is the center and LP passes through O), so the central angle ∠LOP is a straight angle, which measures \( 180^\circ \).

Step2: Length of LP

LP is a diameter. The radius (e.g., OP) is 8.5, so diameter \( LP = 2 \times 8.5 = 17 \) units.

Step3: Length of MP

Triangle LMP is a right triangle (∠LMP is right angle) with LM = 8 and LP = 17. Using Pythagorean theorem: \( MP = \sqrt{LP^2 - LM^2} = \sqrt{17^2 - 8^2} = \sqrt{289 - 64} = \sqrt{225} = 15 \) units.

Answer:

The measure of central angle LOP is \( 180^\circ \).
The length of LP is 17 units.
The length of MP is 15 units.