Question

the measure of central angle lop is \\(\square^{\circ}\\). the length of lp is \\(\square\\) units. the length of mp is \\(\square\\) units. (diagram: circle with center o, points l, m, p on circumference; lm = 8, right angle at m, segment from o to p labeled 8.5, lp is a diameter.)

Explanation:

Step1: Determine central angle LOP

Since \( LM \perp MP \) and \( LP \) is a diameter (as \( O \) is the center and \( LP \) passes through \( O \)), triangle \( LMP \) is a right triangle with \( LP \) as hypotenuse. Also, \( LM = 8 \), \( OP = OL = \frac{LP}{2} \) (radii), and \( OP = 8.5 \)? Wait, no, \( LP \) is the diameter, so \( OL = OP = r \), and \( LP = 2r \). Wait, in the diagram, \( OM \) is a radius? Wait, \( M \) is on the circle, so \( OM \) is a radius. Wait, \( LM = 8 \), \( \angle LMP = 90^\circ \), so \( LP \) is the diameter (since angle inscribed in a semicircle is right angle). So \( LP = 2 \times OM \)? Wait, no, \( O \) is the center, so \( OL = OM = OP = r \). Wait, the length from \( O \) to \( P \) is 8.5? Wait, the diagram shows \( OP = 8.5 \)? Wait, no, the segment from \( O \) to \( P \) is labeled 8.5? Wait, no, the segment from \( O \) to \( P \) is a radius? Wait, no, \( LP \) is the diameter, so \( LP = 2 \times r \). Wait, maybe \( OP = 8.5 \) is the radius? No, \( LP \) is the diameter, so \( LP = 2 \times OP \) if \( O \) is the center. Wait, maybe I misread. Wait, the triangle \( LMP \) is right-angled at \( M \), so \( LP \) is the hypotenuse (diameter), so \( LP = 2 \times OM \) (since \( OM \) is radius). Wait, but \( LM = 8 \), \( \angle LMP = 90^\circ \), so by Pythagoras, \( LP^2 = LM^2 + MP^2 \). Also, \( LP \) is the diameter, so \( LP = 2 \times r \), and \( OM = r \). Wait, maybe \( OP = 8.5 \) is the radius? No, \( OP \) is a radius, so \( LP = 2 \times 8.5 = 17 \)? Wait, that makes sense. So \( LP = 17 \) (diameter), \( OM = 8.5 \) (radius). Then, in right triangle \( LMP \), \( LM = 8 \), \( LP = 17 \), so \( MP = \sqrt{LP^2 - LM^2} = \sqrt{17^2 - 8^2} = \sqrt{289 - 64} = \sqrt{225} = 15 \). Also, since \( LP \) is a diameter, the central angle \( \angle LOP \) is \( 180^\circ \) (since a diameter subtends a straight angle at the center).

Step2: Calculate length of LP

Since \( O \) is the center and \( LP \) is a diameter, \( LP = 2 \times OP \). Wait, but in the diagram, the segment from \( O \) to \( P \) is 8.5? Wait, no, maybe \( OP = 8.5 \) is the radius? Wait, no, \( LP \) is the diameter, so \( LP = 2 \times 8.5 = 17 \). Wait, that's correct because \( OM \) is a radius, and \( LM = 8 \), \( \angle LMP = 90^\circ \), so \( LP \) must be the hypotenuse (diameter), so \( LP = 17 \).

Step3: Calculate length of MP

Using Pythagorean theorem in right triangle \( LMP \): \( MP = \sqrt{LP^2 - LM^2} \). Substituting \( LP = 17 \) and \( LM = 8 \), we get \( MP = \sqrt{17^2 - 8^2} = \sqrt{289 - 64} = \sqrt{225} = 15 \).

Answer:

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