Circles

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\(\begin{aligned}&\text{In the figure shown,}\overline{JM}\text{ is the diameter and }\overline{LM}\text{ is a radius of semicircle }L,\text{ and point }K\text{ is}\\&\text{the midpoint of }\overline{JL}.\text{ If the length of }\overline{KL}\text{ is }13\text{ units and the length of }\overline{MN}\text{ is }14\text{ units, which}\\&\text{expression represents the area, in square units, of this figure?}\end{aligned}\)

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

\(\begin{aligned}
&\text{A circle in the }xy\text{-plane has center }(5, 7)\text{ and radius 2. Which of the following is}\\
&\text{an equation of the circle?}\end{aligned}\)

\(\text{What is the radius of the circle in the }xy\text{-plane defined by }(x+4)^2+(y+8)^2=169?\)

\(\begin{aligned}&\text{What is the diameter of the circle in the }xy\text{-plane with equation }\\
&(x-5)^2+(y-3)^2=16?\end{aligned}\)

\(\begin{aligned}&\text{What is the diameter of the circle in the }xy\text{-plane with equation}\\&(x-5)^2+(y-4)^2=64~?\end{aligned}\)

\(\begin{aligned}&\text{What is the center of the circle in the }xy\text{-plane defined by the equation }\\&(x-9)^{2}+(y+6)^{2}=81?\end{aligned}\)

\(\begin{aligned}
&\text{A circle in the }xy\text{-plane has the equation }(x-13)^{2}+(y-k)^{2}=64.\text{ Which of the following}\\
&\text{gives the center of the circle and its radius?}
\end{aligned}\)

\(\begin{aligned}
&\text{A circle in the }xy\text{-plane has its center at }(16,17)\text{ and has a radius of }7k.\text{ Which equation}\\
&\text{represents this circle?}\end{aligned}\)

\(\begin{aligned}
&\text{A circle in the }xy\text{-plane has its center at }(5, 1)\text{ and a radius with a length of 7. The equation }\\
&\text{for this circle can be written in the form }(x-h)^2+(y-k)^2=b \text{ where }h,~k\text{ and }b\text{ are }\\
&\text{constants. What is the value of }h?\end{aligned}\)

\(\begin{aligned}
&\text{A circle in the }xy\text{-plane has a diameter with endpoints }(2,4) \text{ and }(2,14).\text{ An equation }\\
&\text{of this circle is }\left(x-2\right)^{2}+\left(y-9\right)^{2}=r^{2},\text{where }r\text{ is a positive constant. What is the}\\&\text{value of }r?\end{aligned}\)

\(\begin{aligned}
&\text{The point }(0,0)\text{ lies on a circle in the }xy\text{-plane. An equation of this circle is }x^2+(y+6)^2=p+3\\
&\text{where }p\text{ is a positive constant. What is the value of }p?\end{aligned}\)

\(\begin{aligned}
&\text{In the }xy\text{-plane, the graph of the equation }(x-6)^{2}+(y-2)^{2}=36\text{ is a circle. The point }\\
&(12,c),\text{ where }c\text{ is constant, lies on the circle. What is the value of the }c?\end{aligned}\)

\(\begin{aligned}
&\text{Which of the following is an equation of a circle in the }xy\text{-plane with center }(0,4)\text{ and a }\\&\text{radius with endpoint}\left(\frac{4}{3},5\right)?
\end{aligned}\)

\(\begin{aligned}
&\text{A circle in the }xy\text{-plane has its center at }(-5, 2)\text{ and the point }(-9, 5)\text{ lies on the }\\
&\text{circle.Which equation represents this circle?}\end{aligned}\)

\(\text{The circle shown has its center at }(0,0).\text{ What is the value of }k?\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(x-6)^{2}+(y+5)^{2}=16\\
&\text{In the }xy\text{-plane, the graph of the equation above is a circle. Point }P\text{ is on the circle and }\\
&\text{has coordinates }(10,-5).\text{ If }\overline{PQ}\text{ is a diameter of the circle, what are the coordinates of}\\&\text{point }Q?\end{aligned}\)

\(\begin{aligned}
&\text{A circle in the }xy\text{-plane has a diameter with endpoints }(a,11)\text{ and }(a, d),\text{ where }a\\
&\text{and }d\text{ are constants. An equation of this circle is }(x-a)^{2}+(y-17)^{2}=r^{2},\text{ where }\\
&r\text{ is a positive }\text{constant. What is the value of }d?\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(x+4)^{2}+(y-19)^{2}=121\\
&\text{The graph of the given equation is a circle in the }xy\text{-plane. The point }(a,b)\text{ lies on the circle.}\\&\text{Which of the following is a possible value for }a?\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(x+3)^{2}+(y-4)^{2}=25\\
&\text{In the }xy\text{-plane, the graph of the given equation is a circle. Which point lies on this circle?}
\end{aligned}\)

\(\begin{aligned}
&\text{A circle in the }xy\text{-plane has equation }(x+3)^{2}+(y-1)^{2}=25.\text{ Which of the following}\\
&\text{points does NOT lie in the interior of the circle?}\end{aligned}\)

\(\begin{aligned}
&\text{Which of the following equations represents a circle in the }xy\text{-plane that intersects the }y\text{-axis}\\&\text{at exactly one point?}
\end{aligned}\)

\(\begin{aligned}&\text{Circle A has equation }(x-7)^2+(y+3)^2=1.\text{ In the }xy\text{-plane, circle B is obtained by}\\&\text{translating circle A to the right 4 units.Which equation represents circle B?}\end{aligned}\)

\(\begin{aligned}
&\text{The equation }x^{2}+(y-1)^{2}=49\text{ represents circle A. Circle B is obtained by shifting circle A }\\
&\text{down 2 units in the }xy\text{-plane. Which of the following equations represents circle B?}\end{aligned}\)

\(\begin{aligned}&\text{Circle }F\text{ in the }xy\text{-plane is represented by the equation }(x-13)^2+(y-9)^2=196.\text{ Circle }G\\
&\text{is obtained by shifting circle }F\text{ 5 units to the left and 11 units up. An equation representing}\\&\text{circle }G\text{ is }(x+h)^2+(y+k)^2=196,\text{where }h\text{ and }k\text{ are constants. What is the value of}\\&h+k?\end{aligned}\)

\(
\begin{aligned}
&\text{In the }xy\text{-plane, circle M is the graph of the equation }(x+6a)^2+(y-38a)^2=36a^2,\\
&\text{where }a\text{ is a positive constant. Circle V is obtained by shifting circle M 12a units to the}\\
&\text{right. Which equation represents circle V?}\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{Circle }A: (x-7)^{2}+(y-p)^{2}=21\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{Circle }B:(x+7)^2+(y-p)^2=21\\
&\text{In the given equations, }p\text{ is a positive constant. Which statement correctly compares the }\\
&\text{graphs of circles }A\text{ and }B\text{ in the }xy\text{-plane?}\end{aligned}\)

\(\begin{aligned}&\text{In the }xy\text{-plane, an equation of circle A is }(x-2)^{2}+(y-3)^{2}=9.\text{ Circle B has the same}\\
&\text{center as circle A but has a radius that is twice the radius of circle A. Which equation }\\&\text{represents circle B?}\\\end{aligned}\)

\(\begin{aligned}&\text{Circle A shown is defined by the equation }x^2+(y-5)^2=7.\text{ Circle B (not shown) has the}\\&\text{same radius but is translated 91 units to the right. If the equation of circle B is}\\&(x-h)^2+(y-k)^2=a\text{ , where }h,~k\text{, and }a\text{ are constants, what is the value of }4a?\end{aligned}\)

\(\begin{aligned}
&\text{Circle A in the }xy\text{-plane has the equation }(x+2)^{2}+(y-2)^{2}=9.\text{ Circle B has the same}\\
&\text{center as the circle A. The radius of circle B is two times the radius of circle A. The equation }\\
&\text{defining circle B in the }xy\text{-plane is }(x+2)^{2}+(y-2)^{2}=k,\mathrm{~where~}k\text{ is a constant. What is}\\&\text{the value of }k?\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(x-3)^2+(y+7)^2=25\\&\text{The given equation represents circle }P\text{ in the }xy\text{-plane. Circle }Q\text{ has a center that is 1 unit}\\&\text{to the right of and 2 units below the center of circle }P. \text{ Circle }Q\text{ has a diameter that is}\\&\text{double the diameter of circle }P.\text{ Which equation represents circle }Q?\end{aligned}\)

\(\begin{aligned}
&\text{Circle }A\text{ (shown) is defined by the equation }(x+2)^{2}+y^{2}=9.\text{ Circle }B\text{ (not shown) }\\
&\text{is the result of shifting circle A down 6 units and increasing the radius so that the radius }\\
&\text{of circle B is 2 times the radius of circle }A.\text{ Which equation defines circle }B?\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x^{2}+14x+y^{2}=6y+109\\
&\text{In the }xy\text{-plane, the graph of the given equation is a circle. What is the length of }\\&\text{the circle’s radius?}\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x^{2}+y^{2}+4x-2y=-1\\
&\text{The equation of a circle in the }xy\text{-plane is shown above. What is the radius of the circle?}\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x^{2}+20x+y^{2}+16y=-20\\
&\text{The equation above defines a circle in the }xy\text{-plane. What are the coordinates of the center }\\
&\text{of the circle?}\end{aligned}\)

\(\begin{aligned}&\text{The graph of }x^2+x+y^2+y=\frac{199}{2}\text{ in the }xy\text{-plane}\text{ is a circle. }\\&\text{What is the length of the circle’s radius?}\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x^{2}-6x+y^{2}-12y-36=0\\
&\text{In the }xy\text{-plane, the graph of the given equation is a circle. If this circle is inscribed in }\\
&\text{a square, what is the perimeter of the square?}\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x^{2}-10x+y^{2}-6y-47=0\\
&\text{In the }xy\text{-plane, the graph of the given equation is a circle. If this circle is inscribed in }\\
&\text{a square, what is the perimeter of the square?}\end{aligned}\)

\(\begin{aligned}&\text{A circle in the }xy\text{-plane has its center at }(-4,4)\text{ and has a radius of 6. An equation of this }\\
&\text{circle is }x^2+y^2+ax+by+c=0,\text{ where }a,~b,\text{ and }c\text{ are constants. What is the value of }c?\end{aligned}\)

\(\begin{aligned}
&\text{A circle in the }xy\text{-plane has its center at }(2,-5)\text{ and has a radius of 3. An equation of this }\\
&\text{circle is }x^2+y^2+ax+by+c=0,\text{ where }a,~b,\text{ and }c\text{ are constants. What is the value of }c?\end{aligned}\)

\(\begin{aligned}&\text{In the }xy\text{-plane, the graph of }2x^{2}-6x+2y^{2}+2y=45\text{ is a circle. What is the}\\
&\text{radius of the circle?}\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~2x^2+2y^2+tx+ty-78=0\\
&\text{The given equation, where }t\text{ is a positive constant, defines a circle in the }xy\text{-}\\&\text{plane. The radius of this circle is }\sqrt{41}.\text{ What is the value of }t?\end{aligned}\)

\(\begin{aligned}
&\text{The measure of angle }S\text{ is }\frac{4\pi}{11}\text{ radians. The measure of angle }T\text{ is 3 times the measure of angle }S.\\
&\text{Which expression represents the measure, in }\textbf {degrees},\text{ of angle }T?\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x^{2}+y^{2}-6x-10y-4n=0\\
&\text{The given equation represents circle A in the }xy\text{-plane, where }n\text{ is a constant. Point }\\
&(5,7)\text{ lies on circle B, which has the same center but twice the diameter as circle A.}\\&\text{What is the value of }n?\end{aligned}\)

\(\begin{aligned}
&\text{The number of radians in a 720-degree angle can be written as }a\pi,\text{where }a\text{ is a constant. }\\
&\text{What is the value of }a?\end{aligned}\)

\(\begin{aligned}
&\text{The difference between the measure of angle }A\text{ and the measure of angle }B\text{ is }-\frac{3}{4}\pi\\
&\text{radians. Which expression shows the difference between the measure of angle }A\text{ and}\\&\text{the measure of angle }B,\text{ in degrees?}\\
\end{aligned}\)

\(\begin{aligned}
&\text{The measure of angle }R\text{ is }\frac{2\pi}{3}\text{ radians. The measure of angle }T\text{ is }\frac{5\pi}{12}\text{ radians greater}\\
&\text{than the measure of angle } R.\text{ What is the measure of angle }T,\text{ in }\underline{\text{degrees}}?\end{aligned}\)

\(\begin{aligned}
&\text{The measure of angle }G\text{ is }\frac{\pi}{10}\text{ radians if the measure of angle }G\text{ is }9n\text{ degrees, where }\\&n\text{ is a constant, what is the value of }n?\end{aligned}\)

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

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

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

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

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