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Circle Equations

Grade 10 · Geometry · Worksheet 1

  1. (x + 7)² + (y - 3)² = 121 Answer: ______________
  2. (x + 5)² + (y - 3)² = 49 Answer: ______________
  3. Ava is creating a circular garden in her backyard. She marks the center of the garden at point (6, -1) and wants the edge of the garden to pass through the point (11, 4). What is the radius of Ava's circular garden? Answer: ______________
  4. (x - 9)² + (y + 4)² = 144 Answer: ______________
  5. Liam is designing a circular garden with a center at (5, -10). The garden's boundary passes through the point (15, -10). What is the radius of the garden in meters? Answer: ______________
  6. (x - 12)² + (y + 7)² = 169 Answer: ______________
  7. Find the center and radius of the circle: (x - 5)² + (y + 10)² = 225 Answer: ______________
  8. (x - 9)² + (y - 4)² = 121 Answer: ______________
  9. (x - 9)² + (y - 7)² = 121 Answer: ______________
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Answer Key & Explanations

Circle Equations · Grade 10 · Worksheet 1

  1. (x + 7)² + (y - 3)² = 121 Answer: Center: (-7, 3), Radius: 11 Solution: The standard form of a circle equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Compare (x + 7)² + (y - 3)² = 121 to the standard form.
    Full step-by-step solution

    Step 1: The standard form of a circle equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Step 2: Compare (x + 7)² + (y - 3)² = 121 to the standard form. Step 3: (x + 7)² can be written as (x - (-7))², so h = -7. Step 4: (y - 3)² gives k = 3. Step 5: The right side is 121, which equals r², so r = √121 = 11. Step 6: Therefore, the center is (-7, 3) and the radius is 11.

  2. (x + 5)² + (y - 3)² = 49 Answer: (-5, 3), 7 Solution: The standard form of a circle equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Compare (x + 5)² + (y - 3)² = 49 to the standard form.
    Full step-by-step solution

    Step 1: The standard form of a circle equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Step 2: Compare (x + 5)² + (y - 3)² = 49 to the standard form. Step 3: (x + 5)² can be written as (x - (-5))², so h = -5. Step 4: (y - 3)² gives k = 3. Step 5: The right side is 49, which equals r², so r = √49 = 7. Step 6: Therefore, the center is (-5, 3) and the radius is 7.

  3. Ava is creating a circular garden in her backyard. She marks the center of the garden at point (6, -1) and wants the edge of the garden to pass through the point (11, 4). What is the radius of Ava's circular garden? Answer: 5 Solution: The center of the circle is at (6, -1) A point on the circle is (11, 4) Use the distance formula: radius = sqrt((11 - 6)² + (4 - (-1))²) Calculate: radius = sqrt(5² + 5²) Simplify: radius = sqrt(25 + 25) = sqrt(50) Simplify: sqrt(50) = sqrt(25 × 2) = 5 × sqrt(2) The radius of Ava's circular…
    Full step-by-step solution

    Step 1: The center of the circle is at (6, -1) Step 2: A point on the circle is (11, 4) Step 3: Use the distance formula: radius = sqrt((11 - 6)² + (4 - (-1))²) Step 4: Calculate: radius = sqrt(5² + 5²) Step 5: Simplify: radius = sqrt(25 + 25) = sqrt(50) Step 6: Simplify: sqrt(50) = sqrt(25 × 2) = 5 × sqrt(2) The radius of Ava's circular garden is 5√2.

  4. (x - 9)² + (y + 4)² = 144 Answer: Center: (9, -4), Radius: 12 Solution: The standard form of a circle equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Compare (x - 9)² + (y + 4)² = 144 with the standard form. (x - 9)² means h = 9.
    Full step-by-step solution

    Step 1: The standard form of a circle equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Step 2: Compare (x - 9)² + (y + 4)² = 144 with the standard form. Step 3: (x - 9)² means h = 9. Step 4: (y + 4)² can be written as (y - (-4))², so k = -4. Step 5: The right side is 144, which equals r², so r² = 144. Step 6: Take the square root of both sides: r = √144 = 12. Step 7: Therefore, the center is (9, -4) and the radius is 12.

  5. Liam is designing a circular garden with a center at (5, -10). The garden's boundary passes through the point (15, -10). What is the radius of the garden in meters? Answer: 10 Solution: The center of the circle is at (5, -10) and a point on the circle is (15, -10). The radius is the distance between the center and any point on the circle.
    Full step-by-step solution

    Step 1: The center of the circle is at (5, -10) and a point on the circle is (15, -10). Step 2: The radius is the distance between the center and any point on the circle. Step 3: Use the distance formula: radius = sqrt((15 - 5)^2 + (-10 - (-10))^2) Step 4: Calculate: radius = sqrt((10)^2 + (0)^2) = sqrt(100 + 0) = sqrt(100) Step 5: sqrt(100) = 10 The radius of the garden is 10 meters.

  6. (x - 12)² + (y + 7)² = 169 Answer: Center: (12, -7), Radius: 13 Solution: The standard form of a circle equation is (x - h)² + (y - k)² = r², where (h,k) is the center and r is the radius. Compare (x - 12)² + (y + 7)² = 169 to the standard form. The x-coordinate of the center is h = 12.
    Full step-by-step solution

    Step 1: The standard form of a circle equation is (x - h)² + (y - k)² = r², where (h,k) is the center and r is the radius. Step 2: Compare (x - 12)² + (y + 7)² = 169 to the standard form. Step 3: The x-coordinate of the center is h = 12. Step 4: The y-coordinate of the center is k = -7 (since y + 7 = y - (-7)). Step 5: The radius squared is r² = 169, so r = √169 = 13. Step 6: Therefore, the center is (12, -7) and the radius is 13.

  7. Find the center and radius of the circle: (x - 5)² + (y + 10)² = 225 Answer: Center: (5, -10), Radius: 15 Solution: The standard form of a circle equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Compare (x - 5)² + (y + 10)² = 225 to the standard form.
    Full step-by-step solution

    Step 1: The standard form of a circle equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Step 2: Compare (x - 5)² + (y + 10)² = 225 to the standard form. Step 3: Identify h = 5 and k = -10 (since y + 10 = y - (-10)). So the center is (5, -10). Step 4: Identify r² = 225, so r = sqrt(225) = 15. Step 5: Therefore, the center is (5, -10) and the radius is 15.

  8. (x - 9)² + (y - 4)² = 121 Answer: Center: (9, 4), Radius: 11 Solution: The standard form of a circle equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Compare (x - 9)² + (y - 4)² = 121 with the standard form.
    Full step-by-step solution

    Step 1: The standard form of a circle equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Step 2: Compare (x - 9)² + (y - 4)² = 121 with the standard form. Step 3: Identify h = 9, k = 4, so the center is (9, 4). Step 4: Identify r² = 121, so r = √121 = 11. Step 5: The center is (9, 4) and the radius is 11.

  9. (x - 9)² + (y - 7)² = 121 Answer: Center: (9, 7), Radius: 11 Solution: The standard form of a circle equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Compare (x - 9)² + (y - 7)² = 121 with the standard form.
    Full step-by-step solution

    Step 1: The standard form of a circle equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Step 2: Compare (x - 9)² + (y - 7)² = 121 with the standard form. Step 3: Identify h = 9 and k = 7, so the center is (9, 7). Step 4: Identify r² = 121, so r = √121 = 11. Step 5: Therefore, the center is (9, 7) and the radius is 11.