Drag and Drop
Values should be placed as shown.
You are told that angle L P K \( \angle{LPK} \) is a right angle, which means its measure is pi over 2 \( \frac{\pi}{2} \) radians. That gives the length of arc J K \( \overset{\LARGE\frown}{JK} \) directly on a unit circle.
You are also told that the measure of angle H P J equals one half the measure of angle J P K. \( m\angle{HPJ}=\frac{1}{2}m\angle{JPK}. \) Knowing that the measure of angle J P K \( m\angle{JPK} \) is pi over 2, \( \frac{\pi}{2}, \) the measure of angle H P J \( m\angle{HPJ} \) is one-half that measure, or pi over 4. \( \frac{\pi}{4}. \)
Then, given that the measure of angle L P K equals the measure of angle H P J plus the measure of angle J P K. \( m\angle{LPK}=m\angle{HPJ}+m\angle{JPK}. \) Substituting the values for the measures of angle H P J angle J P K, \( m\angle{HPJ} \textrm{ and } m\angle{JPK}, \) into that equation, you can calculate thatthe measure of angle L P K equals pi over 4 plus pi over 2, which equals 3 pi over 4.
$$ m\angle{LPK}=\frac{\pi}{4}+\frac{\pi}{2} = \frac{3\pi}{4}. $$
Finally, to determine the length of arc L M, \( \overset{\LARGE\frown}{LM}, \) you have to use the fact that the circumference of the entire circle, a unit circle, is 2 pi. \( 2\pi. \) Knowing this and all the arc lengths you have so far (plus the measure of angle M P H equals pi over 3), \( m\angle{MPH} = \frac{\pi}{3}), \) you have the following computation for the length of arc L M. \( \overset{\LARGE\frown}{LM}. \) 2 pi minus open parenthesis measure of angle M P H plus measure of angle H P J plus measure of angle J P K plus measure of angle L P K close parenthesis
$$ 2\pi - ( m\angle{MPH} + m\angle{HPJ} + m\angle{JPK} + m\angle{LPK} ) $$
equals 2 pi minus open parenthesis pi over 3 plus pi over 4 plus pi over 2 plus 3 pi over 4 close parenthesis $$ = 2\pi - ( \frac{\pi}{3} + \frac{\pi}{4} + \frac{\pi}{2} + \frac{3\pi}{4} ) $$
equals 2 pi minus 22 pi over 12 equals 24 pi over 12 minus 22 pi over 12 equals pi over 6 $$ = 2\pi - \frac{22\pi}{12} = \frac{24\pi}{12} - \frac{22\pi}{12} = \frac{\pi}{6} $$
