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Distance and Midpoint

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

Coordinate geometry uses algebra to measure figures. The distance formula comes from the Pythagorean theorem, and the midpoint averages x-values and y-values.

What you'll learn

Why it matters: Navigation apps and robotics use coordinate distance and midpoint ideas to measure paths and choose halfway points.

Worked example

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

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

Answer: (5, 3)

Practice problems

1. Find the distance between (2, 3) and (8, 3).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the distance between (2, 3) and (8, 3).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Same y-value, so use horizontal distance.
  4. Check the result by substituting or estimating: the response should match 6 and make sense in the original problem.

Answer: 6

2. Find the distance between (-1, 4) and (-1, 10).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the distance between (-1, 4) and (-1, 10).
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Same x-value, so use vertical distance.
  4. Check the result by substituting or estimating: the response should match 6 and make sense in the original problem.

Answer: 6

3. Find the midpoint of (0, 0) and (6, 8).

Choices: (3, 4) · (6, 8) · (2, 4) · (4, 3)

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the midpoint of (0, 0) and (6, 8).
  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: (3, 4)

4. 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 the 3-4-5 right triangle.
  4. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

5. Find the distance between (1, 2) and (7, 10).

Show solution
  1. Challenge: First identify exactly what the question is asking: Find the distance between (1, 2) and (7, 10).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Change in x is 6 and change in y is 8.
  4. Distance is sqrt(36 + 64) = 10.
  5. Check the result by substituting or estimating: the response should match 10 and make sense in the original problem.

Answer: 10

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