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Circle Equations

Grade 10 · Geometry · Worksheet 3

  1. (x - 8)² + (y + 6)² = 100 Answer: ______________
  2. (x - 8)² + (y + 9)² = 121 Answer: ______________
  3. Matiu is designing a circular garden for his school. He marks the center of the garden at point (4, -2) on his coordinate grid. If the garden's boundary passes through the point (10, 4), what is the equation of the circle that represents Matiu's garden in standard form?
    • A. (x - 4)² + (y + 2)² = 72
    • B. (x + 4)² + (y - 2)² = 72
    • C. (x - 4)² + (y + 2)² = 36
    • D. (x + 4)² + (y - 2)² = 36
  4. Sophia is creating a circular garden in her backyard. She marks the center of the garden at coordinates (7, -4) relative to her house. If the garden's boundary extends exactly 9 meters from the center, what is the radius of Sophia's garden? Answer: ______________
  5. Find the center and radius of the circle: (x - 7)² + (y + 5)² = 49 Answer: ______________
  6. Liam is designing a circular garden for his backyard. The garden's boundary follows the equation x² + y² + 10x - 20y + 100 = 0. What is the radius of Liam's garden in meters? Answer: ______________
  7. Emma is designing a circular garden for her school project. She marks the center of the garden at point (5, -3) on her coordinate grid. If the garden's boundary passes through the point (11, 1), which equation represents the circular garden?
    • A. (x - 5)² + (y + 3)² = 52
    • B. (x - 5)² + (y + 3)² = 100
    • C. (x + 5)² + (y - 3)² = 52
    • D. (x + 5)² + (y - 3)² = 100
  8. Find the center and radius of the circle: (x - 7)² + (y + 8)² = 81 Answer: ______________
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Answer Key & Explanations

Circle Equations · Grade 10 · Worksheet 3

  1. (x - 8)² + (y + 6)² = 100 Answer: Center: (8, -6), Radius: 10 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 - 8)² + (y + 6)² = 100 with the standard form. For the x-term: (x - 8)² means h = 8.
    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 - 8)² + (y + 6)² = 100 with the standard form. Step 3: For the x-term: (x - 8)² means h = 8. Step 4: For the y-term: (y + 6)² = (y - (-6))² means k = -6. Step 5: The right side is 100, which equals r², so r = √100 = 10. Step 6: Therefore, the center is (8, -6) and the radius is 10.

  2. (x - 8)² + (y + 9)² = 121 Answer: Center: (8, -9), 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 - 8)² + (y + 9)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 - 8)² + (y + 9)² = 121 with the standard form. Step 3: Identify h = 8 and k = -9 (since y + 9 = y - (-9)). Step 4: Identify r² = 121, so r = √121 = 11. Step 5: The center is (8, -9) and the radius is 11.

  3. Matiu is designing a circular garden for his school. He marks the center of the garden at point (4, -2) on his coordinate grid. If the garden's boundary passes through the point (10, 4), what is the equation of the circle that represents Matiu's garden in standard form? Answer: C. (x - 4)² + (y + 2)² = 36 Solution: Identify the center of the circle from the problem: (4, -2) Use the distance formula to find the radius between center (4, -2) and point on circle (10, 4) Distance = sqrt((10 - 4)² + (4 - (-2))²) = sqrt(6² + 6²) = sqrt(36 + 36) = sqrt(72) The radius squared is r² = (sqrt(72))² = 72 Write the…
    Full step-by-step solution

    Step 1: Identify the center of the circle from the problem: (4, -2) Step 2: Use the distance formula to find the radius between center (4, -2) and point on circle (10, 4) Step 3: Distance = sqrt((10 - 4)² + (4 - (-2))²) = sqrt(6² + 6²) = sqrt(36 + 36) = sqrt(72) Step 4: The radius squared is r² = (sqrt(72))² = 72 Step 5: Write the equation in standard form: (x - 4)² + (y - (-2))² = 72 Step 6: Simplify: (x - 4)² + (y + 2)² = 72 Step 7: Compare with answer choices - the correct equation is (x - 4)² + (y + 2)² = 72 Step 8: This corresponds to choice A

  4. Sophia is creating a circular garden in her backyard. She marks the center of the garden at coordinates (7, -4) relative to her house. If the garden's boundary extends exactly 9 meters from the center, what is the radius of Sophia's garden? Answer: 9 Solution: - Center of the circle: (7, -4) - Boundary extends exactly 9 meters from the center - The radius of a circle is the distance from the center to any point on the boundary - Since the boundary extends 9 meters from the center, this distance is the radius - The radius of Sophia's garden is 9 meters
    Full step-by-step solution

    Step 1: Identify what information is given in the problem - Center of the circle: (7, -4) - Boundary extends exactly 9 meters from the center Step 2: Understand what the radius represents - The radius of a circle is the distance from the center to any point on the boundary Step 3: Apply the definition to the given information - Since the boundary extends 9 meters from the center, this distance is the radius Step 4: State the final answer - The radius of Sophia's garden is 9 meters

  5. Find the center and radius of the circle: (x - 7)² + (y + 5)² = 49 Answer: Center: (7, -5), Radius: 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 - 7)² + (y + 5)² = 49 to the standard form. The term (x - 7) means h = 7.
    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 + 5)² = 49 to the standard form. Step 3: The term (x - 7) means h = 7. Step 4: The term (y + 5) can be written as (y - (-5)), so k = -5. Step 5: The right side of the equation is 49, which is r². So, r² = 49. Step 6: Taking the positive square root, r = √49 = 7. Step 7: Therefore, the center is (7, -5) and the radius is 7.

  6. Liam is designing a circular garden for his backyard. The garden's boundary follows the equation x² + y² + 10x - 20y + 100 = 0. What is the radius of Liam's garden in meters? Answer: 5 Solution: Start with the equation: x² + y² + 10x - 20y + 100 = 0 Group x terms and y terms: (x² + 10x) + (y² - 20y) = -100 Complete the square for x: (10/2)² = 25, so add 25 to both sides Complete the square for y: (-20/2)² = 100, so add 100 to both sides The equation becomes: (x² + 10x + 25) + (y² - 20y…
    Full step-by-step solution

    Step 1: Start with the equation: x² + y² + 10x - 20y + 100 = 0 Step 2: Group x terms and y terms: (x² + 10x) + (y² - 20y) = -100 Step 3: Complete the square for x: (10/2)² = 25, so add 25 to both sides Step 4: Complete the square for y: (-20/2)² = 100, so add 100 to both sides Step 5: The equation becomes: (x² + 10x + 25) + (y² - 20y + 100) = -100 + 25 + 100 Step 6: Factor the perfect squares: (x + 5)² + (y - 10)² = 25 Step 7: The standard form is (x - h)² + (y - k)² = r², so r² = 25 Step 8: Therefore, r = sqrt(25) = 5 The radius of Liam's garden is 5 meters.

  7. Emma is designing a circular garden for her school project. She marks the center of the garden at point (5, -3) on her coordinate grid. If the garden's boundary passes through the point (11, 1), which equation represents the circular garden? Answer: A. (x - 5)² + (y + 3)² = 52 Solution: Identify the center coordinates from the problem: (5, -3) Identify the point on the circle: (11, 1) Calculate the radius using the distance formula: r = sqrt((11 - 5)² + (1 - (-3))²) Calculate the differences: (11 - 5) = 6 and (1 - (-3)) = 4 Square the differences: 6² = 36 and 4² = 16 Add the…
    Full step-by-step solution

    Step 1: Identify the center coordinates from the problem: (5, -3) Step 2: Identify the point on the circle: (11, 1) Step 3: Calculate the radius using the distance formula: r = sqrt((11 - 5)² + (1 - (-3))²) Step 4: Calculate the differences: (11 - 5) = 6 and (1 - (-3)) = 4 Step 5: Square the differences: 6² = 36 and 4² = 16 Step 6: Add the squares: 36 + 16 = 52 Step 7: The radius squared is r² = 52 (we don't need to take the square root since the equation uses r²) Step 8: Write the equation in standard form: (x - 5)² + (y - (-3))² = 52, which simplifies to (x - 5)² + (y + 3)² = 52 Step 9: Compare with the options - this matches option C The correct answer is (x - 5)² + (y + 3)² = 52.

  8. Find the center and radius of the circle: (x - 7)² + (y + 8)² = 81 Answer: Center: (7, -8), Radius: 9 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 + 8)² = 81 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 + 8)² = 81 to the standard form. Step 3: The x-coordinate of the center is h = 7 (from x - 7). Step 4: The y-coordinate of the center is k = -8 (from y + 8, which is y - (-8)). Step 5: The radius squared is r² = 81, so r = √81 = 9. Step 6: Therefore, the center is (7, -8) and the radius is 9.