Circles - PSAT Math

1. State the chord-chord power theorem: chords intersect at $E$ with segments $AE,EB$ and $CE,ED$.

   $AE\cdot EB=CE\cdot ED$. Products of chord segments are equal at intersection.

2. Find the arc length when $r=6$ and $\theta=60^\circ$.

   $2\pi$. $s=\frac{60}{360}\cdot 2\pi(6)=\frac{1}{6}\cdot 12\pi=2\pi$.

3. State the equation of a circle with center $(h,k)$ and radius $r$ in standard form.

   $(x-h)^2+(y-k)^2=r^2$. Standard form shows center coordinates and radius squared.

4. What is the relationship between diameter $d$ and radius $r$ in a circle?

   $d=2r$. Diameter is twice the radius.

5. Find the distance between the centers of two circles with centers $(1,2)$ and $(4,6)$.

   $5$. Distance formula: $\sqrt{(4-1)^2+(6-2)^2}=\sqrt{9+16}=5$.

6. What is the measure of an inscribed angle that intercepts an arc of measure $m$ degrees?

   $\frac{m}{2}$. Inscribed angle theorem: half the intercepted arc.

7. State the area formula for a circle with radius $r$.

   $A=\pi r^2$. Area equals $\pi$ times radius squared.

8. Identify the radius of the circle $(x+1)^2+(y-4)^2=49$.

   $7$. Radius equals the square root of the constant term.

9. Identify the center and radius of $x^2+y^2-8x+6y=0$.

   Center $(4,-3)$, radius $5$. Complete the square: $(x-4)^2+(y+3)^2=25$.

10. Find the arc length for $r=6$ and $\theta=60^\circ$.

    $2\pi$. $s=\frac{60}{360}\cdot 2\pi(6)=\frac{1}{6}\cdot 12\pi=2\pi$.

11. What is the length of a semicircle arc with radius $r$ (arc only, not including the diameter)?

    $\pi r$. Half the circumference gives the semicircle arc.

12. Find the measure of an inscribed angle that intercepts a $110^\circ$ arc.

    $55^\circ$. Inscribed angle is half the intercepted arc: $\frac{110°}{2}=55°$.

13. Find the radius if the center is $(2,3)$ and a point on the circle is $(2,11)$.

    $8$. Distance from $(2,3)$ to $(2,11)$ is $|11-3|=8$.

14. State the standard form equation of a circle with center $(h,k)$ and radius $r$.

    $(x-h)^2+(y-k)^2=r^2$. Standard form shows center coordinates and radius squared.

15. Find the radius of the circle $(x+5)^2+(y-7)^2=49$.

    $7$. $r^2=49$, so $r=\sqrt{49}=7$.

16. Use $PT^2=PA\cdot PB$: if $PA=4$ and $PB=16$, find $PT$.

    $8$. $PT^2=4\cdot 16=64$, so $PT=\sqrt{64}=8$.

17. What is the diameter of a circle in terms of radius $r$?

    $d=2r$. Diameter is twice the radius.

18. State the tangent-secant power theorem from external point $P$: tangent $PT$, secant meets circle at $A$ and $B$.

    $PT^2=PA\cdot PB$. Tangent squared equals product of secant segments from P.

19. What is the circumference formula of a circle with radius $r$?

    $C=2\pi r$. Circumference equals $2\pi$ times the radius.

20. State the standard form of a circle with center $(h,k)$ and radius $r$.

    $(x-h)^2+(y-k)^2=r^2$. Standard form shows center and radius directly.

21. State the circumference formula for a circle with radius $r$.

    $C=2\pi r$. Circumference equals $2\pi$ times the radius.

22. What are the center and radius of $(x-3)^2+(y+2)^2=25$?

    Center $(3,-2)$, radius $5$. Read $(h,k)$ from $(x-h)^2+(y-k)^2$ and $r$ from $r^2=25$.

23. What is the arc length formula for central angle $\theta$ in degrees and radius $r$?

    $s=\frac{\theta}{360}\cdot 2\pi r$. Arc length is the fraction of circumference.

24. Identify the center and radius from $(x-3)^2+(y+4)^2=25$.

    Center $(3,-4)$, radius $5$. Read $(h,k)$ from $(x-h)^2+(y-k)^2$ and $r=\sqrt{25}=5$.

25. Find the center of the circle $(x+5)^2+(y-7)^2=49$.

    $(-5,7)$. Center is $(h,k)$ where equation is $(x-h)^2+(y-k)^2$.

26. State the relationship between diameter $d$ and radius $r$.

    $d=2r$. Diameter is twice the radius.

27. What is the area formula of a circle with radius $r$?

    $A=\pi r^2$. Area equals $\pi$ times radius squared.

28. What is the radius of a circle in terms of diameter $d$?

    $r=\frac{d}{2}$. Radius is half the diameter.

29. What is the distance formula used to find a radius from center $(h,k)$ to point $(x,y)$?

    $r=\sqrt{(x-h)^2+(y-k)^2}$. Distance formula gives radius from center to any point.

30. What is the measure of a central angle that intercepts an arc of measure $m$ degrees?

    $m$. Central angle equals its intercepted arc measure.

31. Find the area of a sector for $r=3$ and $\theta=120^\circ$.

    $3\pi$. $A=\frac{120}{360}\cdot \pi(3)^2=\frac{1}{3}\cdot 9\pi=3\pi$.

32. State the sector area formula for central angle $\theta$ (in degrees) and radius $r$.

    $A=\frac{\theta}{360}\cdot \pi r^2$. Fraction of full area based on angle ratio.

33. State the equation of a circle centered at the origin with radius $r$.

    $x^2+y^2=r^2$. Center at origin means $(h,k)=(0,0)$.

34. State the arc length formula for central angle $\theta$ in degrees and radius $r$.

    $s=\frac{\theta}{360}\cdot 2\pi r$. Arc length is the fraction of circumference for given angle.

35. Use $AE\cdot EB=CE\cdot ED$: if $AE=3$, $EB=12$, and $CE=6$, find $ED$.

    $6$. $3\cdot 12=6\cdot ED$, so $ED=\frac{36}{6}=6$.

36. Find the arc length of a $90^\circ$ sector in a circle of radius $8$.

    $4\pi$. Use $s=\frac{90}{360}\cdot 2\pi(8)=\frac{1}{4}\cdot 16\pi$.

37. Which statement is always true about a radius drawn to a point of tangency?

    It is perpendicular to the tangent line. Radius meets tangent at $90°$ angle.

38. Identify the center and radius of $(x-3)^2+(y+2)^2=25$.

    Center $(3,-2)$, radius $5$. Read $(h,k)$ from $(x-h)^2+(y-k)^2$; $r^2=25$ so $r=5$.

39. State the circumference of a circle with diameter $d$.

    $C=\pi d$. Circumference equals $\pi$ times diameter.

40. Find the area of a circle with diameter $10$.

    $25\pi$. Use $A=\pi r^2$ with $r=5$ (half of diameter).

41. State the relationship between a tangent and the radius at the point of tangency.

    Tangent is perpendicular to the radius. Forms a $90°$ angle at the point of tangency.

42. Find the area of a sector when $r=10$ and $\theta=90^\circ$.

    $25\pi$. $A=\frac{90}{360}\cdot \pi(10)^2=\frac{1}{4}\cdot 100\pi=25\pi$.

43. What is the sector area formula for central angle $\theta$ in degrees and radius $r$?

    $A=\frac{\theta}{360}\cdot \pi r^2$. Sector area is the fraction of circle area.

44. State the tangent property relating a radius to a tangent line at the point of tangency.

    A radius is perpendicular to the tangent: $r\perp \text{tangent}$. Radius meets tangent at $90°$ angle.

45. Identify the radius of $(x+1)^2+(y-2)^2=\frac{49}{4}$.

    $\frac{7}{2}$. $r^2=\frac{49}{4}$, so $r=\frac{7}{2}$.

46. State the circle equation with center $(2,-1)$ and radius $4$.

    $(x-2)^2+(y+1)^2=16$. Substitute center $(h,k)=(2,-1)$ and $r^2=16$.

47. What is the length of a semicircle arc (not including the diameter) with radius $r$?

    $\pi r$. Half the circumference, excluding the diameter.

48. State the arc length formula for central angle $\theta$ (in degrees) and radius $r$.

    $s=\frac{\theta}{360}\cdot 2\pi r$. Fraction of full circumference based on angle ratio.

49. Find the circumference of a circle with diameter $12$.

    $12\pi$. Use $C=2\pi r$ with $r=6$ (half of diameter).

50. Find the area of a $60^\circ$ sector of a circle with radius $6$.

    $6\pi$. Use $A=\frac{60}{360}\cdot\pi(6)^2=\frac{1}{6}\cdot 36\pi$.

51. State the formula for the circumference of a circle with radius $r$.

    $C=2\pi r$. Circumference equals $2\pi$ times the radius.

52. State the formula for the area of a circle with radius $r$.

    $A=\pi r^2$. Area equals $\pi$ times radius squared.

53. State the sector area formula for central angle $\theta$ in degrees and radius $r$.

    $A=\frac{\theta}{360}\cdot \pi r^2$. Sector area is the fraction of circle area for given angle.

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