Question

The circle in the figure above has center O. Which of the following measures for the figure would be sufficient by itself to determine the radius of the circle?
I. The length of arc PQR
II. The perimeter of triangle OPR
III. The length of chord PR.
see the attachment for figure. please show all your work

Answer 1

You could determine radius using any of I,II, or III
I:
Circumference = 2pi*r
arc PQR is 1/4 circumference
\[\rightarrow r = \frac{4*PQR}{2\pi}\]
II:
Perimeter is sum of 3 sides of triangle, and 2 sides equal the radius
isosceles right triangle implies 45-45-90 triangle of ratio 1:1:sqrt(2)
P =2r +sqrt(2)r
\[\rightarrow r = \frac{P}{2+\sqrt{2}}\]

III:
chord PR is known as sqrt(2)*r from II
\[\rightarrow r=\frac{PR}{\sqrt{2}}\]

Answer 2

thank u