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SAT Math Circle Equations: Geometry & Coordinate Shifts

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SAT Math Circle Equations: Geometry & Coordinate Shifts

Master Standard Form Equations and Geometric Shifts for Digital SAT Math Success

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Geometry accounts for approximately 15% of all questions on the Digital SAT Math section. Among these, coordinate geometry questions involving circles frequently appear on both Module 1 and Module 2. To score high, you need to quickly extract the center and radius from the standard circle equation and know how geometric shifts (translations) alter the – and -coordinates of the center without changing the circle’s size. Today, we are putting both skills to the test!
 

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📌 Core Formula: Standard Equation of a Circle

Before tackling today’s practice problems, commit this fundamental formula to memory:

(x – h)2 + (y – k)2 = r2

  • (h, k) represents the center of the circle. Notice the subtraction signs: if you see (x – 4), h = +4; if you see (x + 4), h = – 4 .
  • r represents the radius. The right side of the equation equals r2, so you must take the square root of that constant to find r.

✏️ Practice Questions: Test Your Skills

📝 QUESTION 47: Equation of a Circle

An equation of a circle in the -plane is given below:

(x – 4)2 + (y + 7)2 = 36

Which of the following statements correctly describes the center and the radius of the circle?

  • A) The center is at (4, – 7) and the radius is 6.
  • B) The center is at (– 4, 7) and the radius is 6.
  • C) The center is at (4, – 7) and the radius is 36.
  • D) The center is at (– 4, 7) and the radius is 36.

📝 QUESTION 48: Translating a Circle (Geometric Shifts)

A circle in the xy-plane is defined by the equation (x + 1)2 + (y – 3)2 = 16. If the circle is translated 4 units to the right and 5 units down, which equation represents the new circle?

  • A)     (x + 5)2 + (y – 8)2 = 16
  • B)     (x – 3)2 + (y + 2)2 = 16
  • C)     (x – 3)2 + (y + 2)2 = 4
  • D)     (x – 5)2 + (y + 2)2 = 16

📝 QUESTION 45: Cylinder Volume

A cylindrical water bottle has a radius of 4 cm and a height of 15 cm. What is the volume of the water bottle in cubic centimeters?

  • A) 60 cm³
  • B) 120 cm³
  • C) 240 cm³
  • D) 480 cm³

🔍 Step-by-Step Solution

  1. Identify the given values and formula:

Volume (V) = πr2h

  • Given radius (r): 4 cm
  • Given height (h): 15 cm
  1. Plug the values into the formula:

V = π (4)2 (15)

  1. Simplify exponent and multiply:

V = π (16) (15) = 240π cm3

Correct Answer: C (240π cm3)

📝 QUESTION 46: Solving for Radius

A cylindrical soup can has a volume of 300 cm3 and a height of 12 cm. What is the approximate radius of the soup can?

  • A)     2.8 cm
  • B)     3.2 cm
  • C)     4.5 cm
  • D)     5.6 cn

🔍 Step-by-Step Solution

  1. Set up the volume formula:

Volume (V) = πr2h

  1. Substitute the known values:

300 = πr²(12)

  1. Isolate r2 by dividing both sides by 12π:

r2 = 300/12π = 25/π

  1. Calculate the value using π ≈14159:

r2 ≈ 25/3.14159 ≈ 7.9577

  1. Take the square root of both sides:

r = √(7.9577) ≈ 2.82 cm

Rounding to the nearest tenth gives approximately 2.8 cm

Correct Answer: A (2.8 cm)

 

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