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Distance and midpoint formula worksheets

Ten questions a sheet, a new set at the press of a button, and an answer key with the working.

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Same sheet number, same questions.

The distance formula

Name:

Find the distance between each pair of points. Give the exact answer in simplest radical form, then round it to 2 decimal places.

d = √((x₂ − x₁)² + (y₂ − y₁)²)

  1. A(−7, 3) and B(1, 9)d =
  2. A(−1, 7) and B(−6, 7)d =
  3. A(−9, 0) and B(−6, −4)d =
  4. A(−7, 10) and B(−3, 2)d =
  5. A(6, −10) and B(1, 2)d =
  6. A(−4, 5) and B(4, 5)d =
  7. A(4, 5) and B(−4, −10)d =
  8. A(−8, −4) and B(1, −3)d =
  9. A(−2, −9) and B(3, 3)d =
  10. A(−8, 10) and B(−2, 4)d =
Made at stringmash.com/distance-formula-worksheet · sheet 1

The distance formula: answer key

  1. A(−7, 3) and B(1, 9) √(8² + 6²) = √(64 + 36) = √100 = 10
  2. A(−1, 7) and B(−6, 7) √((−5)² + 0²) = √(25 + 0) = √25 = 5
  3. A(−9, 0) and B(−6, −4) √(3² + (−4)²) = √(9 + 16) = √25 = 5
  4. A(−7, 10) and B(−3, 2) √(4² + (−8)²) = √(16 + 64) = √80 = 4√5 ≈ 8.94
  5. A(6, −10) and B(1, 2) √((−5)² + 12²) = √(25 + 144) = √169 = 13
  6. A(−4, 5) and B(4, 5) √(8² + 0²) = √(64 + 0) = √64 = 8
  7. A(4, 5) and B(−4, −10) √((−8)² + (−15)²) = √(64 + 225) = √289 = 17
  8. A(−8, −4) and B(1, −3) √(9² + 1²) = √(81 + 1) = √82 ≈ 9.06
  9. A(−2, −9) and B(3, 3) √(5² + 12²) = √(25 + 144) = √169 = 13
  10. A(−8, 10) and B(−2, 4) √(6² + (−6)²) = √(36 + 36) = √72 = 6√2 ≈ 8.49
Made at stringmash.com/distance-formula-worksheet · sheet 1

What's on the sheets

Each sheet has ten pairs of points with whole-number coordinates from −10 to 10, the formula at the top and room to work under each question. Pick the questions:

Distance: find the distance between each pair. Half the questions are built on Pythagorean triples such as 3, 4, 5 and 8, 15, 17, so the answer is a whole number. The other half give roots to simplify, like √50 = 5√2, then round to 2 decimal places.

Midpoint and endpoint: eight midpoints, some landing on halves, then two questions that give one end and the midpoint and ask for the other end.

Distance and midpoint: both for each pair, for revision or a quiz.

The answer key prints on a second page with the working for each question, so it can be used for marking or handed back as worked examples. The sheet number picks the questions: the same number always gives the same sheet, so a class can share one by number.

The two formulas

The distance between (x₁, y₁) and (x₂, y₂) is d = √((x₂ − x₁)² + (y₂ − y₁)²). It's the Pythagorean theorem on the right triangle with one leg across and one leg up between the points.

The midpoint is M = ((x₁ + x₂)/2, (y₁ + y₂)/2), the average of the two points. Run backwards, the missing endpoint is B = 2M − A, one coordinate at a time.

Distance between two points on a grid is Common Core standard 8.G.B.8 in grade 8. In high school geometry, standard G-GPE.B.6 asks for the point that divides a segment in a given ratio, and the midpoint is the 1 : 1 case. Standard G-GPE.B.7 uses the distance formula to find perimeters.

Questions

Are the answers exact?

Yes. Distances are given in simplest radical form, such as 2√2, with the decimal to 2 places. Midpoints are given as fractions and decimals, such as (13/2, 8) = (6.5, 8).

Can I check a single answer?

Type the two points into the [distance formula calculator](/distance-formula-calculator) or the [midpoint calculator](/midpoint-calculator). They use the same working as the answer key.

Does it print on A4?

Yes. Pick A4 or US Letter before printing. Untick the answer key to print the questions only.

Sources

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