Geometry Workbook — Chapter 15: Equations of Circles

SOL G.PC.4 (a, b) · Companion to Textbook Chapter 15

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 112.

PAGE 1 — Chapter opener

Chapter 15 · Equations of Circles

Standard G.PC.4 (a, b)

In this chapter you will:

• Derive the standard equation of a circle from the Pythagorean Theorem
• Write a circle's equation directly from its center and radius
• Read a center and a radius off an equation — and off a graph
• Determine a circle's equation from the endpoints of a diameter
• Determine a circle's equation from its center and one point on the circle
• Decide which of those four starting points a problem has handed you

Words to know: standard form · center · radius · Pythagorean Theorem · leg · hypotenuse · diameter · midpoint · simplest radical form

Conventions: standard form is $(x-h)^2+(y-k)^2=r^2$. Both terms are subtractions — a negative center coordinate flips the sign you see. The right side is $r^2$, not $r$. Radii are given in simplest radical form, never a rounded decimal.

PAGE 2 — The whole chapter on one page

One Triangle, Four Problem Types

Every row ends at a center and a radius — the only two numbers a circle's equation ever needs.

Before every problem:

• Am I reading (equation or graph → center, radius) or writing (two pieces of information → equation)?
• If writing: do I already have the center, or do I need to find it first?

PAGE 3 — The right triangle inside every circle

15.1 Where the Equation Comes From

FIGURE: fig1-deriving-the-equation.png (full width)

Fill in from the figure.

Horizontal leg: ____________________ Vertical leg: ____________________ Hypotenuse: ____________________

$$(x - \_\_)^2 + (y - \_\_)^2 = \_\_^2$$

PAGE 4 — The origin case

15.1 When the Center Is (0, 0)

FIGURE: fig2-circle-at-the-origin.png (full width)

$$x^2 + y^2 = r^2$$

Not a different formula — the $h = 0$, $k = 0$ case of the same one.

PAGE 5 — Writing from center and radius

15.1 No Point to Measure To

FIGURE: fig6-writing-from-center-and-radius.png (full width)

Copy the three steps, in order.

Step one: ____________________ Step two: ____________________ Step three: ____________________

PAGE 6 — Guided practice 15.1

Work Through It

1. Name the two legs of the right triangle and what each one measures. ______
2. Name the hypotenuse and give the equation it produces. ______
3. Give the equation shown and identify $h$ and $k$. ______
4. Explain why a center at the origin shortens the equation. ______
5. Give the center and the radius shown. ______
6. Give the three steps that turn a center and a radius into an equation. ______

PAGE 7 — Write it

15.1 Center and Radius to Equation

Give the equation.

7. center $(0, 0)$, radius $3$ ______
8. center $(0, 0)$, radius $8$ ______
9. center $(4, 0)$, radius $5$ ______
10. center $(0, -6)$, radius $6$ ______
11. center $(5, 2)$, radius $7$ ______
12. center $(-3, 1)$, radius $4$ ______
13. center $(-2, -5)$, radius $9$ ______
14. center $(6, -4)$, radius $10$ ______

PAGE 8 — On the circle, or not?

15.1 Check It

Answer yes or no.

15. $(3, 4)$ on $x^2+y^2=25$ ______
16. $(2, 2)$ on $x^2+y^2=9$ ______
17. $(7, 1)$ on $(x-3)^2+(y-1)^2=16$ ______
18. $(5, 5)$ on $(x-1)^2+(y-2)^2=25$ ______
19. $(0, 0)$ on $(x-3)^2+(y-4)^2=25$ ______
20. $(1, 1)$ on $(x+2)^2+(y-3)^2=20$ ______

PAGE 9 — Derive it

15.1 Center and a Point

Derive the equation.

21. center $(0, 0)$, point $(6, 8)$ ______

22. center $(1, 1)$, point $(4, 5)$ ______

23. center $(-2, 3)$, point $(1, -1)$ ______

24. Application. A fountain is centered at the origin of a park's grid, in meters, and its edge reaches $(9, 12)$. Give its equation.

25. Error analysis. A student derives center $(2, -3)$, radius $5$ as $(x+2)^2+(y-3)^2=25$. Identify and correct the sign errors.

26. Reasoning. Explain why the right side is $r^2$, not $r$, tracing back to the right triangle.

Exit ticket 15.1

27. Give the equation with center $(0, 5)$ and radius $3$. ______
28. Is $(4, 3)$ on $x^2+y^2=25$? Justify using the Pythagorean Theorem. ______

PAGE 10 — Anatomy of the equation

15.2 Every Equation Has This Shape

FIGURE: fig3-anatomy-of-the-equation.png (full width)

Fill in.

The number subtracted from $x$ is ______. The number subtracted from $y$ is ______.

A negative center coordinate makes the sign you see ____________________.

The number on the right is ______, not $r$ — finding the radius costs one ____________________.

PAGE 11 — A table of them

15.2 Read the Equation, Write Down Two Things

FIGURE: fig4-reading-center-and-radius.png (full width)

PAGE 12 — Reading a graph

15.2 No Equation — Just the Picture

FIGURE: fig5-reading-a-graph.png (full width)

The radius is easiest to read straight up, down, left, or right from the center — where the circle crosses a grid line exactly.

PAGE 13 — When the radius is a radical

15.2 Not Every Radius Is a Whole Number

FIGURE: fig9-radius-in-radical-form.png (full width)

$$r^2 = \_\_^2 + \_\_^2 = \_\_ \;\rightarrow\; r = \_\_\_\_$$

PAGE 14 — Guided practice 15.2

Work Through It

29. Give the center and the radius shown. ______
30. Explain why the center's negative $x$-coordinate makes the equation show a plus sign. ______
31. Give the center and the radius directly from the picture. ______
32. Say which directions are easiest to measure a radius along, and why. ______
33. Give the equation, center, and radius from the radical row of the table. ______
34. Give the two legs, $r^2$, and the radius in simplest radical form. ______

PAGE 15 — Read it

15.2 Equation to Center and Radius

Give the center and radius.

35. $(x-4)^2+(y-1)^2=36$ ______
36. $(x+2)^2+(y-5)^2=49$ ______
37. $(x-6)^2+(y+3)^2=64$ ______
38. $x^2+(y-7)^2=81$ ______
39. $(x+5)^2+y^2=100$ ______
40. $(x-1)^2+(y+8)^2=144$ ______
41. $(x+9)^2+(y+2)^2=169$ ______
42. $x^2+y^2=121$ ______

PAGE 16 — Radical radii

15.2 Simplest Radical Form

Give the center and radius.

43. $(x-2)^2+(y-1)^2=18$ ______
44. $(x+1)^2+(y-4)^2=20$ ______
45. $x^2+(y+3)^2=50$ ______
46. $(x-3)^2+y^2=45$ ______

PAGE 17 — Graphs, and back to writing

15.2 Both Directions

47. Center $(2, -3)$, through $(2, 4)$ straight up. Give $r$ and the equation. ______
48. Center $(-5, 1)$, through $(1, 1)$ straight across. Give $r$ and the equation. ______
49. Center $(0, 0)$, through $(0, -9)$. Give $r$ and the equation. ______

Give the equation.

50. center $(7, -5)$, radius $2\sqrt3$ ______

51. center $(-6, -6)$, radius $15$ ______

52. Application. A cell tower at $(5, -2)$ on a town's grid, in miles, has signal reaching $(5, 6)$. Confirm $r$ and give the equation.

53. Error analysis. A student reads $(x-4)^2+(y+2)^2=49$ as center $(4, 2)$, radius $49$. Identify both mistakes.

54. Reasoning. Explain why finding $r$ from an equation needs a square root but finding the center does not.

Exit ticket 15.2

55. Give the center and radius of $(x+3)^2+(y-6)^2=40$, radius in simplest radical form. ______
56. Center $(1, 4)$, through $(1, -2)$. Give $r$ and the equation. ______

PAGE 18 — Center and a point on the circle

15.3 One Point Pins Down the Radius

FIGURE: fig7-center-and-a-point.png (full width)

$$r^2 = \_\_^2+\_\_^2=\_\_ \;\rightarrow\; r=\_\_$$

PAGE 19 — The endpoints of a diameter

15.3 The Circle in Disguise

FIGURE: fig8-diameter-endpoints.png (full width)

• center = ____________________ of the two endpoints
• radius = distance from the center to ____________________
• not the distance ____________________, which is the diameter

PAGE 20 — Guided practice 15.3

Work Through It

57. Give the two legs of the triangle and the radius they produce. ______
58. Give the resulting equation. ______
59. Give the center's coordinates, and explain how you found them. ______
60. Give the radius, and name the segment you measured. ______
61. Explain what goes wrong if the full diameter is used as the radius. ______
62. Compare the two methods: what step is shared, and what step is different? ______

PAGE 21 — Center and a point

15.3 Write the Equation

63. Center $(3, 5)$, point $(7, 8)$ ______
64. Center $(-2, 4)$, point $(10, -1)$ ______
65. Center $(0, 0)$, point $(-8, 6)$ ______
66. Center $(1, -3)$, point $(1, 5)$ ______
67. Center $(4, -1)$, point $(-2, 7)$ ______
68. Center $(-5, 2)$, point $(3, -4)$ ______
69. Center $(0, 3)$, point $(5, 15)$ ______
70. Center $(6, 6)$, point $(-3, -6)$ ______

PAGE 22 — Diameter endpoints

15.3 Write the Equation

Endpoints of a diameter. Give the equation.

71. $A(0, 0)$, $B(6, 8)$ ______

72. $A(-4, 1)$, $B(4, 7)$ ______

73. $A(2, -5)$, $B(2, 9)$ ______

74. $A(-6, -2)$, $B(2, 4)$ ______

75. $A(1, 3)$, $B(9, 9)$ ______

76. $A(-3, 7)$, $B(5, -3)$ ______

77. $A(-8, -4)$, $B(4, 4)$ ______

78. $A(2, 2)$, $B(2, -14)$ ______

79. $A(-1, -1)$, $B(7, 5)$ ______

80. Application. A lighthouse beam centers on $(4, -3)$, kilometers, and reaches $(4, 9)$. Give the equation.

81. Error analysis. Diameter endpoints $A(2, 6)$, $B(10, 2)$ — a student adds instead of averages to find the center. Give the correct center and equation.

82. Reasoning. Both methods end at a center and a radius. Which does each skip, and which does it compute?

Exit ticket 15.3

83. Center $(5, 5)$, point $(9, 8)$ on the circle. ______
84. $A(0, 6)$, $B(8, 0)$ endpoints of a diameter. ______

PAGE 23 — Four starting points, two computations

15.4 The Whole Chapter, One Table

FIGURE: fig10-the-four-types.png (full width)

PAGE 24 — The most common mistake

15.4 Watch the Sign

FIGURE: fig11-the-sign-error.png (full width)

PAGE 25 — What each part controls

15.4 Same Center or Same Radius

FIGURE: fig12-two-circles-compared.png (full width)

PAGE 26 — In context

15.4 Three Real Circles

FIGURE: fig13-equations-in-context.png (full width)

PAGE 27 — Guided practice 15.4

Work Through It

85. Give the "how" column for each of the four given-information types. ______
86. Give the wrong equation, the correct one, and what changed. ______
87. Give both equations in the left panel and what changed. ______
88. Give both equations in the right panel and what changed. ______
89. Name which of the four types each context panel is. ______
90. Give one blank panel's center and radius by reading the grid. ______

PAGE 28 — Name it, then solve

15.4 Which Type Is This?

Name the type, then solve.

91. $(x-5)^2+(y+2)^2=81$ ______
92. $(x+7)^2+(y-3)^2=144$ ______
93. $x^2+(y+9)^2=100$ ______
94. $(x-8)^2+(y-8)^2=169$ ______
95. Graphed: center $(3, -5)$, through $(3, 3)$ straight up. ______
96. Graphed: center $(-6, 0)$, through $(0, 0)$. ______
97. Diameter $A(0, 4)$, $B(8, -2)$. ______
98. Diameter $A(-5, -5)$, $B(3, 1)$. ______
99. Center $(2, -6)$, point $(10, 0)$. ______
100. Center $(-4, 4)$, point $(-4, -10)$. ______

PAGE 29 — More mixed practice

15.4 Keep Going

101. Diameter $A(-8, -2)$, $B(0, 4)$. ______

102. Center $(0, 0)$, point $(3, -3)$ — radical radius. ______

103. $(x-4)^2+(y+1)^2=50$ ______

104. $(x+2)^2+(y+3)^2=72$ ______

105. Application. A weather balloon station at the origin, km, loses signal at $(8, 15)$. Give the equation.

106. Application. A plaza's diameter is staked at $(-8, 3)$ and $(0, -3)$, meters. Give the equation.

107. Application. A lighthouse beam centers on $(3, -6)$, km, and reaches $(3, 2)$. Give the equation.

108. Error analysis. Given $(x+1)^2+(y-5)^2=40$, a student states $r=40$. Give the correct radius.

109. Reasoning. Why are a midpoint and a segment length genuinely different computations?

110. Reasoning. For $(x-a)^2+(y-b)^2=c$ to be an actual circle, what must be true of $c$?

Exit ticket 15.4

111. Center $(-3, 5)$, point $(3, -3)$ on the circle. ______
112. State the one question to ask before choosing among the four methods. ______

PAGE 30 — Blank circles

Your Turn

FIGURE: fig14-blank-circle-equation-frames.png (full width)

For every problem in this chapter:

• Decide: am I reading (equation/graph → center, radius) or writing (two facts → equation)?
• If writing, find the center first if it isn't already given — everything else follows from a center and a radius.
• Watch the sign: $(x-h)$ with a negative $h$ prints as a plus.
• Give radii in simplest radical form unless a decimal is asked for.

PAGE 31 — Chapter review

Chapter 15 Review

Review 1 (G.PC.4a). A circle has center $C(-4, 2)$ and passes through $P(2, 10)$.

• Give the horizontal and vertical legs the derivation uses.
• Use the Pythagorean Theorem to find $r^2$ and $r$.
• Give the equation of the circle.
• Check whether $R(-10, -4)$ is also on the circle, and explain how you know without fully re-deriving.

Review 2 (G.PC.4b). A circle's equation is $(x-6)^2+(y+9)^2=169$.

• Give the center and the radius.
• A second circle is graphed with center $(-4, 5)$, through $(-4, -3)$ straight below. Give its radius and equation.
• Compare the two radii — which is bigger, and by how much?
• Explain why finding the radius needed a square root but finding the center did not.

Review 3 (G.PC.4b). $A(-7, -4)$ and $B(1, 2)$ are diameter endpoints of one circle. A second circle has center $(-3, -8)$ and passes through $(9, -3)$.

• Give the equation of the first circle.
• Give the equation of the second circle.
• Name the method used for each.
• Say which single quantity both methods still had to compute, even though they started from different information.