25 hard SAT Circles questions

A circle has center O and radius 12. Points P and Q lie on the circle. The length of arc PQ is $8\pi$. What is the measure, in degrees, of central angle POQ?

• A$60$
• B$120$
• C$240$
• D$8$

In the xy-plane, a circle has the equation $x^{2} + y^{2} - 14x + 6y + 42 = 0$. What is the circumference of the circle?

• A$4\pi$
• B$8\pi$
• C$16\pi$
• D$8$

Circle A has equation $(x + 3)^{2} + (y - 4)^{2} = 16$. Circle B is the image of circle A under a translation 5 units left and 2 units down, followed by a dilation by a factor of 3 centered at the origin. Which equation represents circle B?

• A$(x + 24)^{2} + (y - 6)^{2} = 144$
• B$(x + 24)^{2} + (y - 6)^{2} = 48$
• C$(x + 8)^{2} + (y - 2)^{2} = 16$
• D$(x + 8)^{2} + (y - 2)^{2} = 144$

A circle has a radius of 8 meters. An arc of the circle has a length of $10\pi$ meters. What is the measure of the angle subtended by the arc at the center of the circle, in degrees?

• A$225$
• B$125$
• C$75$
• D$200$

Circle T has a radius of 10. An inscribed angle intercepts an arc of length $5\pi$. What is the measure, in radians, of this inscribed angle?

• A$\pi /4$
• B$\pi /2$
• C$\pi /8$
• D$\displaystyle \frac{1}{4}$

In the xy-plane, a circle has equation $x^{2} + y^{2} + 10x - 4y - 7 = 0$. A sector of this circle has a central angle of $\pi /3$ radians. What is the area of this sector?

• A$6\pi$
• B$12$
• C$12\pi$
• D$2\pi$

In the xy-plane, a circle has equation $x^{2} + y^{2} - 6x + 8y + 9 = 0$. What is the radius of the circle?

• A$4$
• B$8$
• C$16$
• D$\displaystyle \sqrt{7}$

In the xy-plane, a circle has equation $x^{2} + y^{2} + 4x - 6y - 12 = 0$. A line is tangent to the circle at point (2, m) where m > 0. What is the value of m?

• A$3$
• B$5$
• C$6$
• D$8$

Circle C has center at the origin in the xy-plane. Points P and Q lie on circle C such that the coordinates of P are $(0, k)$ where $k > 0$, and the coordinates of Q are (8, 6). If the length of arc PQ is $5\pi$, what is the value of k?

• A$5$
• B$8$
• C$10$
• D$20$

A circle in the xy-plane has its center at (h, k) and has a radius of 13. The circle passes through the origin and through the point (10, 0). What is the value of k?

• A$0$
• B$5$
• C$12$
• D$13$

Circle A has equation $(x - 4)^{2} + y^{2} = 49$. Circle B is the image of circle A under a translation 2 units right and 6 units down, followed by a dilation by a factor of 3 centered at the origin. Which equation represents circle B?

• A$(x - 6)^{2} + (y + 6)^{2} = 49$
• B$(x - 6)^{2} + (y + 6)^{2} = 441$
• C$(x - 18)^{2} + (y + 18)^{2} = 147$
• D$(x - 18)^{2} + (y + 18)^{2} = 441$

An arc of a circle has a length of $7\pi$ inches and subtends an angle at the center of the circle. The radius of the circle is 3 inches. What is the measure of the angle in degrees?

• A$420$
• B$140$
• C$210$
• D$240$

A circle has center O and radius 24. Points T and U lie on the circle. The measure of inscribed angle TOU, with vertex on the circle at point V, is 25°. What is the measure, in degrees, of central angle TOU?

• A$12.5$
• B$25$
• C$50$
• D$0.436$

A circle has center at the origin in the xy-plane. Point M with coordinates (8, 15) lies on the circle. A sector of this circle has a central angle of $\pi /3$ radians. What is the area of this sector?

• A$289\pi /6$
• B$\displaystyle \frac{289}{6}$
• C$289\pi /3$
• D$17\pi /3$

A circle has center O and points P and Q lie on the circle. The length of arc PQ is $5\pi$ inches and the measure of central angle POQ is $150^{\circ}$. What is the radius, in inches, of the circle?

• A$6$
• B$12$
• C$15$
• D$5\pi /150$

In the xy-plane, a circle passes through the points (0, 0), (8, 0), and (0, 6). What is the equation of the circle in the form $x^{2}$ + $y^{2}$ + ax + by + $c = 0$?

• A$x^{2} + y^{2} - 8x - 6y = 0$
• B$x^{2} + y^{2} - 4x - 3y = 0$
• C$x^{2} + y^{2} - 8x - 6y + 48 = 0$
• D$x^{2} + y^{2} - 16x - 12y = 0$

Circle A has equation $(x - 1)^{2} + (y + 7)^{2} = 9$. Circle B is the image of circle A under a translation 8 units left and 4 units up, followed by a dilation by a factor of 5 centered at the origin. Which equation represents circle B?

• A$(x + 7)^{2} + (y + 3)^{2} = 9$
• B$(x + 7)^{2} + (y + 3)^{2} = 225$
• C$(x + 35)^{2} + (y + 15)^{2} = 45$
• D$(x + 35)^{2} + (y + 15)^{2} = 225$

A circle has center O, and points M and N lie on the circle. An inscribed angle MPQ intercepts arc MN, and the measure of central angle MON is 84°. What is the measure, in degrees, of inscribed angle MPQ?

• A$21$
• B$42$
• C$84$
• D$7\pi /15$

In a circle with center O and radius r, a sector has central angle $\pi /3$ radians and area $24\pi$ square centimeters. What is the value of r?

• A$24$
• B$12$
• C$144$
• D$60$

A circle has center O. Points J and K lie on the circle such that the length of arc JK is $3\pi$ and the radius of the circle is 12. What is the measure, in degrees, of arc JK?

• A$15$
• B$30$
• C$45$
• D$90$

In the xy-plane, a circle has center at (4, -3) and passes through the point (8, 0). If the circle has equation $x^{2}$ + $y^{2}$ + ax + by + $c = 0$, what is the value of c?

• A$-8$
• B$0$
• C$8$
• D$25$

Circle A has equation $x^{2} + (y + 4)^{2} = 36$. Circle B is the image of circle A under a translation 7 units left and 1 unit up, followed by a dilation by a factor of 2 centered at the origin. Which equation represents circle B?

• A$(x + 14)^{2} + (y + 6)^{2} = 72$
• B$(x + 14)^{2} + (y + 6)^{2} = 144$
• C$(x + 7)^{2} + (y + 3)^{2} = 36$
• D$(x + 7)^{2} + (y + 3)^{2} = 144$

A circle has center O, and points M and N lie on the circle. The central angle MON has measure $120^{\circ}$ and the area of sector MON is $12\pi$ square units. What is the radius, in units, of the circle?

• A$6$
• B$12$
• C$36$
• D$3$

In the xy-plane, a circle has center at (4, -6) and passes through the point (7, -2). An equation of this circle is $x^{2}$ + $y^{2}$ + ax + by + $c = 0$, where a, b, and c are constants. What is the value of c?

• A$-27$
• B$5$
• C$25$
• D$27$

A circle in the xy-plane has equation $x^{2} + y^{2} - 6x + 8y + 9 = 0$. What is the radius of the circle?

• A$4$
• B$8$
• C$16$
• D$3$