Answer key: Distance & Midpoint Worksheet

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Goal: Find the distance between two points with the Pythagorean theorem, find midpoints by averaging coordinates, and spot horizontal and vertical segments.

Connects to Common Core 8.G.B.8 (Apply the Pythagorean theorem to find the distance between two points in a coordinate system.)

Completed results table

Values are the readouts of the tool (Distance, midpoint & slope) for each setting.

Answers

1. For A(0, 0) and B(3, 4), use Δx, Δy and the Pythagorean theorem to find the distance.

   Δx = 3 and Δy = 4, so d = √(3² + 4²) = √25 = 5. The distance formula is the Pythagorean theorem on the coordinate grid.

2. Find the distance from A(−3, −2) to B(4, 5) by hand. Write it as a square root, then round.

   Δx = 7 and Δy = 7, so d = √(49 + 49) = √98 ≈ 9.899.

3. How do you find a midpoint without the tool? Show it for A(−4, 3) and B(2, −5).

   Average the x-coordinates and the y-coordinates: ((−4 + 2) ÷ 2, (3 + (−5)) ÷ 2) = (−1, −1).

4. Two rows have a distance of 6. What is special about these pairs of points, and what are their slopes?

   A(1, 2) to B(7, 2) is horizontal: Δy = 0, so the distance is just Δx = 6 and the slope is 0. A(2, −1) to B(2, 5) is vertical: Δx = 0, so the distance is Δy = 6 and the slope is undefined, because you cannot divide by 0.

5. M(4, 2) is the midpoint of segment AB, and A is (1, 2). Find B.

   B is as far past M as A is before it: B = (2 × 4 − 1, 2 × 2 − 2) = (7, 2). The third row of the table confirms it.

6. Which pair of points is farthest apart? Show the distance calculation.

   A(−4, 3) and B(2, −5): Δx = 6 and Δy = −8, so d = √(36 + 64) = √100 = 10, the largest distance in the table.