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Distance, Midpoint, and Circles

A free College Algebra lesson from the “Relating Algebra and Geometry” unit, with a worked example and practice problems including step-by-step solutions.

Coordinate formulas connect algebra and geometry. Distance comes from the Pythagorean theorem, midpoint averages coordinates, and circle equations describe all points a fixed distance from a center.

What you'll learn

Why it matters: Maps, navigation, screen coordinates, and circular boundaries all turn geometry into algebra. Distance, midpoint, and circle equations let students describe location and radius precisely from coordinates.

Worked example

Problem. Find the midpoint of (2, 9) and (8, 1).

  1. Average x-values: (2 + 8)/2 = 5.
  2. Average y-values: (9 + 1)/2 = 5.
  3. The midpoint is (5, 5).

Answer: (5, 5)

Practice problems

1. Find the midpoint of (0, 4) and (6, 10).

Choices: (3, 7) · (6, 14) · (3, 14) · (7, 3)

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the midpoint of (0, 4) and (6, 10).
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Average x and y values.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (3, 7)

2. Find the distance between (0, 0) and (3, 4).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the distance between (0, 0) and (3, 4).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Use a 3-4-5 triangle.
  4. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

3. For (x - 2)^2 + (y + 1)^2 = 25, what is the radius?

Show solution
  1. Challenge: First identify exactly what the question is asking: For (x - 2)^2 + (y + 1)^2 = 25, what is the radius?
  2. For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
  3. r^2 = 25.
  4. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

4. Find the midpoint of (2, 5) and (8, 1).

Choices: (5, 3) · (6, 4) · (10, 6) · (3, 5)

Show solution
  1. Midpoint: First identify exactly what the question is asking: Find the midpoint of (2, 5) and (8, 1).
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Average x-values and y-values.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (5, 3)

5. Find the distance between (-1, -2) and (2, 2).

Show solution
  1. Distance: First identify exactly what the question is asking: Find the distance between (-1, -2) and (2, 2).
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. The changes are 3 and 4.
  4. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

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