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 Question from Angela, a student: Segment PR is a diameter of circle S. If angle P = 25, find minor arc QR. This circle has an isosceles triangle in it, it is connected to the diameter, this question has stumped so bad, I am close to graduating, and I am homeschooled, can't find help anywhere, I hope you can help me?

Angie,

Using the notation in the diagram you sent

triangle PQS is isosceles and the measure of angle SPQ is 25o so the measure of angle PQS is also 25o. The sum of the measures of the angles in a triangle is 180o so the measure of the angle QSP is 180o - (25o + 25o) = 130o. PSR is a line segment so the measure of the angle RSQ is 180o - 130o = 50o. Can you now find the length of the minor arc QR?

If you need further assistance write back,
Harley

Angie wrote back

Harley,

Thanks for trying to help me, I actually was looking for the formula, and this was kind of confusing to me, I still can't figure out the answer, I am even more confused?

Thanks,
Angie

Angie,

I showed you that the measure of angle RSQ is 50o. The sector RSQ is only part of the circle. The entire circle has a central angle of 360o so sector RSQ is a fraction of the circle, in fact the sector is 50/360 = 5/36th of the circle. The entire circumference of the circle has length 2πr where r is the radius and thus the length of the arc QR is 5/36th of this length. Hence the length of the arc QR is

5/36 × 2πr = 5/18 πr.

Harley

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