Question

calculating lengths and measures of angles in circles. the measure of central angle lop is. the length of lp is units. the length of mp is units. 8 15 17 20

Explanation:

Step1: Recall circle - angle property

The measure of an inscribed - angle subtended by a diameter is 90 degrees. Since $\angle LMP = 90^{\circ}$ and $\angle LOP$ is the central angle subtended by the same arc $\overset{\frown}{LP}$, the measure of central angle $\angle LOP$ is twice the measure of inscribed angle $\angle LMP$. So $\angle LOP=180^{\circ}$.

Step2: Use the Pythagorean theorem in right - triangle LMP

In right - triangle $LMP$, let $LM = 8$ and $LP$ is the diameter of the circle with radius $r = 8.5$, so $LP=2r = 17$.

Step3: Apply the Pythagorean theorem to find MP

In right - triangle $LMP$, by the Pythagorean theorem $LP^{2}=LM^{2}+MP^{2}$. We know $LP = 17$ and $LM = 8$. Then $MP=\sqrt{LP^{2}-LM^{2}}=\sqrt{17^{2}-8^{2}}=\sqrt{(17 + 8)(17 - 8)}=\sqrt{25\times9}=\sqrt{225}=15$.

Answer:

The measure of central angle $\angle LOP$ is $180^{\circ}$
The length of $LP$ is $17$ units
The length of $MP$ is $15$ units