Line PR is the diameter of Circle S. S is the middle. If the measure of P = 24 find the measure of arc QR. Please help! Explain the process pleaseeee.
(this sign < means angle)
the triangle PQS is an isoscele triangle (both legs are radii of the circle).
Therefore <SPQ = <SQP.
Using the theorem of the sum of angles in a triangle you get:
<QSR = 2 * 24° = 48°
Therefore the arc is:
arc(QR) = 48° * (2pi)/360° * (1/2)PR = (4/15)pi * (1/2)PR ≈ 0.8377... * PR
Wait...if QP = 48 deg. Isn't QR the difference from 180 degrees? I'm so confused..and no. My teacher never explains anything... *mumbles He's not exactly the best teacher. He gives us the work and expects us to know it all automatically. The whole class is confused. If I understand the problems I share what I learned with my friends and vice-versa. Well, thanks for any help!
i think earboth did a good job of explaining. but let's try to go through his process more slowly. we notice that triangle PQS is an isoseles triangle. an isoseles triangle is a triangle having two sides and two angles equal. since PS and QS are the radii of the circle, they are the same length, and so we have PQS being an isoseles triangle.
now the two angles that are equal are the ones formed by the two equal sides with the other side. so angle PQS is equal to angle QPS.
we are told that angle P = 24 degrees, so angle Q must be 24 degrees as well. since the angle in a triangle add up to 180 degrees, angle PSQ must be 132 degrees. and since the angles on a straight line add up to 180 degrees, angle PSQ plus angle QSR must add up to 180. so,
angle QSR = 180 - angle PSQ = 180 - 132 = 48 degrees.
now this last angle is the angle that subtends the arc we are concerned with. the formula for the length of an arc in degrees is:
length of arc = (theta)/360 * 2pi*r
where theta is the angle that subtends the arc, and r is the radius. since we don't know the radius, earboth used (1/2)PR, since that is the radius in terms of PR. so now he just plugged everything into the formula.
length of QR = 48/360 * 2pi * (1/2)PR
and he simplified that to 0.4189*PR ..............at least he should have simplified to that, he seems to have forgotten about the 1/2 in front of the PR