Circles

\(\begin{aligned}
&\text{Point }O\text{ is the center of a circle. }\text{The measure of arc } RS\text{ on this circle is }100^{\circ}.\text{ What is }\\
&\text{the measure, in degrees, of its associated angle }ROS?
\end{aligned}\)

\(\begin{aligned}
&\text{A circle has a center }O\text{, and points } R\text{ and }S\text{ lie on the circle. In triangle }ORS,\\
&\text{the measure of }\angle {ROS}\text { is }88^{\circ}.\text{What is the measure of }\angle {RSO},\text{ in degrees}?
\end{aligned}\)

\(\text{Point }P\text{ is the center of the circle in the figure above. What is the value of }x?\)

\(\begin{aligned}
&\text{Points }F\text{and }G\text{ lie on a circle with center }H.\text{ Segment }FG\text{ is a diameter of the circle. }\\
&\text{If the length of segment }FG\text{ is }98\text{ centimeters, what is the length, in centimeters, of }\\&\text{segment }FH?\end{aligned}\)

\(\begin{aligned}
&\text{Points }Q\text{ and }R \text { lie on a circle with center }P. \text{ The radius of this circle is 9 inches. }\\
&\text{Triangle }PQR \text{ has a perimeter of 31 inches. What is the length, in inches, of }\overline{QR} ?\end{aligned}\)

\(\begin{aligned}
&\text{A circle in the }xy\text{-plane has its center at }(-2, -4).\text{ Line }k\text{ is tangent to this circle at the point }\\
&(-5,-5),\text{ What is the slope of line }k?\end{aligned}\)

\(\begin{aligned}
&\text{A circle in the }xy\text{-plane has its center at }(3, 7).\text{ Line }t\text{ is tangent to this circle at the point }\\
&(a,-4),\text{ where }a\text{ is a constant. The slope of line }t\text{ is }\frac{5}{4}\text .\text{ What is the value of }a?\end{aligned}\)

\(\begin{aligned}
&\text{A circle in the }xy\text{-plane has its center at }(-1,1).\text{ Line }t\text{ is tangent to this circle at the point }\\
&(5,-4).\text{ Which of the following points also lies on line }t?\end{aligned}\)

\(\begin{aligned}
&\text{A circle in the }xy\text{-plane has its center at }(-2,2).\text{ Line }t\text{ is tangent to this circle at the point }\\
&(7,-5).\text{ Which of the following points also lies on line }t?\end{aligned}\)

\(\begin{aligned}
&\text{A circle has center }G,\text{ and points }M\text{ and }N\text{ lie on the circle. Line segments }MH\\
&\text{and }NH\text{ are tangent to}\text{ the circle at points }M\text{ and }N,\text{respectively. If the radius}\\
&\text{of the circle is }168\text{ millimeters and the perimeter of quadrilateral }GMHN\text{ is}\\
&\text{3,856 millimeters, what is the distance, in millimeters, between points }G\text{ and }H?\end{aligned}\)

\(\begin{aligned}&\text{The perimeter of an equilateral triangle is 852 centimeters. The three vertices of the triangle}\\
&\text{lie on a circle. The radius of the circle is }w\sqrt{3}\text{ centimeters. What is the value of }w?\end{aligned}\)

\(\begin{aligned}
&\text{In the figure, }\overline{AC}\text{ is a diameter of the circle with center }O,\text{ and }AC=10.\text{ Triangle }\\
&\triangle ACB\text{ is equilateral. The length of }\overline{BP}\text{ is }n\text{ times the length of }\overline{AO}.\text{ What is the }\\&\text{value of }n?\end{aligned}\)

\(\begin{aligned}
&\text{In the figure shown, points }A,~B,~C\text{ and }E\text{ lie on the circle, and }AB > BC.\\
&\text{Segment }AC\text{ is perpendicular to segment }BE\text{ at point }D,\text{ and }BD=\sqrt{346}.\text{ The}\\
&\text{diameter of the circle is 175. If }\frac{CD}{AD}=r,\text{ what is the value of }r?
\end{aligned}\)

\(\begin{aligned}
&\text{A square is inscribed in a circle. The radius of the circle is }\frac{20\sqrt{2}}{2}\text{ inches. What is the} \\
&\text{side length, in inches, of the square?}\end{aligned}\)

\(\begin{aligned}&\text{A rectangle is inscribed in a circle, such that each vertex of the rectangle lies on the }\\&\text{circumference of the circle. The diagonal of the rectangle is twice the length of the }\\&\text{shortest side of the rectangle. The area of the rectangle is }1{,}089\sqrt{3}\text{ square units. }\\&\text{What is the length, in units, of the diameter of the circle?}\end{aligned}\)

\(\begin{aligned}
&\text{A circle has a circumference of }42\pi\text{ centimeters. What is the diameter, in centimeters, }\\&\text{of the circle?}\end{aligned}\)

\(\begin{aligned}
&\text{A circle has a radius of 2.1 inches. The area of the circle is }b\pi\text{ square inches, where }b\text{ is a constant.}\\
&\text{What is the value of }b?\end{aligned}\)

\(\begin{aligned}
&\text{Circle }A\text{ has a radius of }3n \text{ and circle }B\text{ has a radius of }129n,\text{ where }n\text{ is a positive}\\
&\text{constant. The area of circle }B\text{ is how many times the area of circle }A?\end{aligned}\)

\(\begin{aligned}
&\text{Circle }A\text{ has a radius of }2x \text{ and circle }B\text{ has a radius of }94x.\text{ The area of circle }B\\
&\text{is how many times the area of circle }A?\end{aligned}\)

\(\begin{aligned}
&\text{Points }M,N,\text{ and }P\text{ lie on the circle shown. On this circle, minor arc }MN\text{ has a length}\\
&\text{of 39 centimeters and major arc }MPN\text{ has a length of 195 centimeters. What is the}\\
&\text{circumference, in centimeters, of the circle shown?}\end{aligned}\)

\(\begin{aligned}
&\text{The circle shown has center }O,\text{ circumference 144}\pi,\text{and diameters }\overline{PR}\text{ and }\overline{QS}.\\
&\text{The length of arc }PS\text{ is twice the length of arc }PQ.\text{ What is the length of arc }QR?\end{aligned}\)

\(\begin{aligned}
&\text{A circle has diameters }\overline{AC}\text{ and }\overline{BD}.\text{ The Cirumference of the circle }84\pi\text{ and the length }\\
&\text{of arc } DA\text{ is 2 times the length of arc }AB.\text{ What is the length of arc }BC?\end{aligned}\)

\(\begin{aligned}&\text{The three points shown define a circle. The circumference of this circle is }k\pi,\text{ where }k\text{ is a}\\&\text{constant. What is the value of }k?\end{aligned}\)

\(\begin{aligned}
&\text{Point }C\text{ is the center of the circle above. What fraction of the area of the circle is the area of}\\
&\text{the shaded region?}\end{aligned}\)

\(\begin{aligned}
&\text{In the circle above, point }A\text{ is the center and the length of arc }{BC}\text{ is }\frac{2}{5}\text{ of the circumference}\\
&\text{of the circle. What is the value of }x?\end{aligned}\)

\(\begin{aligned}
&\text{The circle above with center }O\text{ has a circumference of 36. What is the length }\\
&\text{of minor arc }{AC}?\end{aligned}\)

\(\begin{aligned}
&\text{In the circle above, segment }AB\text{ is a diameter. If the length of arc }{ADB}\text{ is }8\pi,\\
&\text{what is the length of the radius of the circle?}\end{aligned}\)

\(\begin{aligned}&\text{The circle above has center }O,\text{ the length of arc }{ADC}\text{ is }5\pi\text{, and }x=100.\\&\text{What is the length of arc }{ABC}?\end{aligned}\)

\(\begin{aligned}
&\text{A circle has center }P,\text{and points }A\text{ and }B\text{ lie on the circle. The measure of arc }AB\text{ is }45°\\
&\text{and the length of arc }AB\text{ is }4\pi\text{ units. What is the length, in units, of the radius of the circle?}\end{aligned}\)

\(\begin{aligned}
&\text{A circle has center }O,\text{and points }A\text{ and }B\text{ lie on the circle. The measure of arc }AB\text{ is }60°,\\
&\text{and the length of this arc is }4\text{ inches. What is the circumference, in inches, of the circle?}\end{aligned}\)

\(\begin{aligned}
&\text{A circle has center }O,\text{ and a points }A\text{ and }B\text{ lie on the circle. The measure of arc }AB\\
&\text{is }20^{\circ},\text{ and the length of this arc is 4 inches. What is the circumference, in inches, of }\\
&\text{the circle?}\end{aligned}\)

\(\begin{aligned}
&\text{In the figure above, point }O\text{ is the center of the circle, line segments }LM\text{ and }MN\text{ are tangent }\\
&\text{to the circle at points }L\text{ and }N,\text{ respectively, and the segments intersect at point }M\text{ as shown. }\\
&\text{If the circumference of the circle is 96, what is the length of minorarc }{LN}?\end{aligned}\)

\(\begin{aligned}
&\text{Points }A\text{ and }B\text{ lie on a circle with radius }1,\text{ and arc }{AB}\text{ has length }\frac{\pi}{3}.\\
&\text{What fraction of the circumference of the circle is the length of arc }{AB}?\end{aligned}\)

\(\begin{aligned}
&\text{A rectangle is inscribed in a circle such that the length of the diagonal of the rectangle is}\\
&\text{twice the length of its shortest side. The circumference of the circle is }114\pi\text{ units. What is}\\
&\text{the area, in square units, of the rectangle?}\end{aligned}\)

\(\begin{aligned}&\text{A circle is inscribed in a square such that the circumference of the circle touches the}\\
&\text{midpoint of each side of the square. The length of the diagonal of the square is 176}\\
&\text{units. What is the area, in square units, of the circle?}\end{aligned}\)

\(\begin{aligned}&\text{The figure shown consists of a rectangle and a semicircle, where the length of }\overline{WX}\text{ is 20}\\&\text{units and the length of }\overline{WZ}\text{ is 5 units. The diameter of the semicircle is }\overline{WX}.\text{ Which of the}\\&\text{following expressions represents the area, in square units, of the figure?}\end{aligned}\)

\(\begin{aligned}
&\text{Point }F\text{ lies on a unit circle in the }xy\text{-plane and has coordinates }(1,0).\text{ Point }G\text{ is the center}\\&\text{of the circle and has coordinates }(0,0).\text{ Point }H\text{ also lies on the circle and has coordinates}\\&(-1,y).\text{ where }y\text{ is a constant. Which of the following could be the positive measure of angle}\\&FGH,\text{in radians?}\end{aligned}\)

\(\begin{aligned}
&\text{Point }N\text{ lies on a unit circle in the }x\text{y-plane and has coordinates }(1,0).\text{ Point }O\text{ is the}\\
&\text{center of the circle and has coordinates }(0,0).\text{ Point }M\text{ also lies on the unit circle,}\\&\text{and the measure of angle }NOM\text{ is }\frac{272\pi}{3}\text{ radians. If the coordinates of point }M\text{ are}\\&(a,b),\text{ where }a\text{ and }b\text{ are constants, what is the value of }a?\end{aligned}\)

\(\begin{aligned}
&\text{In the }xy\text{-plane, a unit circle with center at the origin }O\text{ contains point }A\text{ with coordinates}\\
&(1,0)\text{ and point }B\text{ with coordinates }\left(\frac{5}{\sqrt{34}},-\frac{3}{\sqrt{34}}\right).\text{ If the measure of angle }AOB\text{ is }p\\&\text{radians, what is the value of
}\frac{\cos p}{\sin p}?\end{aligned}\)

\(\begin{aligned}
&\text{In the }xy\text{-plane, a unit circle with center at the origin }O\text{ contains point }A\text{ with coordinates}\\
&(1,0)\text{ and point }B\text{ with coordinates }\left(-\frac{5}{\sqrt{89}},\frac{8}{\sqrt{89}}\right).\text{ If the measure of angle }AOB\text{ is }w\\&\text{radians, what is the value of
}\tan w?\end{aligned}\)

\(\begin{aligned}
&\text{In the }xy\text{-plane, a unit circle with center at the origin }O\text{ contains point }A\text{ with coordinates}\\
&(0,1).\text{ Point }P\text{ is obtained by rotating point }A~\frac{7\pi}{6}\text{ radians about the origin in the }\\
&\text{counterclockwise direction. What are the coordinates of point }P?
\end{aligned}\)