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Square roots and the Pythagorean theorem

Ten questions of mixed difficulty, covering Square roots and Pythagoras. Print it, or work through it on screen — the answer key starts on its own page.

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Square roots and the Pythagorean theorem

Math 8 · maddyhelps.com

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Circle the best answer for each question. Show your work in the space provided.

  1. √50 lies between which two whole numbers?

    1. a) 25 and 26
    2. b) 7 and 8
    3. c) 6 and 7
    4. d) 49 and 64

  2. Which number is a perfect square?

    1. a) 184
    2. b) 164
    3. c) 144
    4. d) 124

  3. A square has an area of 64 cm². How long is each side?

    1. a) 32 cm
    2. b) 16 cm
    3. c) 8 cm
    4. d) 4096 cm

  4. Which of these products is a perfect square?

    1. a) 2 × 2 × 3 × 3 × 5 × 7
    2. b) 2 × 2 × 3 × 3 × 5 × 5
    3. c) 2 × 2 × 2 × 2 × 3 × 5
    4. d) 2 × 2 × 2 × 3 × 3 × 3

  5. A right triangle has legs of 5 cm and 12 cm. How long is its hypotenuse?

    1. a) 169 cm
    2. b) 7 cm
    3. c) 13 cm
    4. d) 17 cm

  6. A 13 m ladder leans against a wall with its foot 5 m from the bottom of the wall. How high up the wall does the ladder reach?

    1. a) 144 m
    2. b) 13.9 m
    3. c) 8 m
    4. d) 12 m

  7. A square garden has an area of 196 m². How much fencing does it take to go all the way around it?

    1. a) 784 m
    2. b) 56 m
    3. c) 28 m
    4. d) 14 m

  8. To check that a deck's corner is square, Leo marks 60 cm along one edge and 80 cm along the other. The marks are 97 cm apart. What should he conclude?

    1. a) It is not square: a right angle would make that distance 140 cm.
    2. b) It is square: 60, 80 and 97 are close enough to 3, 4 and 5.
    3. c) It is not square: a right angle would make that distance 100 cm.
    4. d) It is square, because 60 + 80 is more than 97.

  9. A rectangular field is 80 m long and 60 m wide. Instead of walking along two sides to the opposite corner, Aisha cuts straight across. How much shorter is her walk?

    1. a) 20 m
    2. b) 100 m
    3. c) 40 m
    4. d) 140 m

  10. Which list is in order from least to greatest?

    1. a) √40, √50, 6, 7
    2. b) 6, √40, √50, 7
    3. c) 6, 7, √40, √50
    4. d) 6, √40, 7, √50

Answer key · Square roots and the Pythagorean theorem

Math 8 · maddyhelps.com

  1. b) 7 and 8 — 7² = 49 and 8² = 64, and 50 is between them, so √50 is between 7 and 8 — just above 7, since 50 is so close to 49. 49 and 64 are the perfect squares on either side, not their roots, and 25 and 26 come from halving 50, but a square root is not half.
  2. c) 144 — 144 = 12 × 12, so a square with an area of 144 square units has whole-number sides. The others also end in 4, as many squares do, but that is not a test: 124 falls between 11² = 121 and 12² = 144, 164 between 12² and 13² = 169, and 184 between 13² and 14² = 196.
  3. c) 8 cm — Each side is the square root of the area, because side × side = area: 8 × 8 = 64, so each side is 8 cm. A square root is not half — a 32 cm side would give an area of 1024 cm² — and 16 cm divides by 4, as if 64 cm were the perimeter; 4096 squares 64 instead of taking its root.
  4. b) 2 × 2 × 3 × 3 × 5 × 5 — The factors of 2 × 2 × 3 × 3 × 5 × 5 split into two identical groups, 2 × 3 × 5 and 2 × 3 × 5, so it equals 30 × 30 = 900. 2 × 2 × 2 × 3 × 3 × 3 has the same number of each factor, but three 2s cannot be shared evenly between two groups — it is 216, between 14² = 196 and 15² = 225. In the other two, some factors have no partner: a 5 and a 7, or a 3 and a 5.
  5. c) 13 cm — Use a² + b² = c²: 5² + 12² = 25 + 144 = 169, so c = √169 = 13 cm. 169 cm stops one step early — 169 is c², not c — and 17 cm adds the sides themselves, but only their squares add up.
  6. d) 12 m — The ladder is the hypotenuse, so h² + 5² = 13²: h² = 169 − 25 = 144 and h = 12 m. 13.9 m adds the squares, √(169 + 25), which makes the height more than the ladder's length — impossible. 144 m forgets to take the square root.
  7. b) 56 m — Each side is √196 = 14 m, because 14 × 14 = 196, and the fence runs along all four sides: 4 × 14 = 56 m. 14 m is one side and 28 m is only two of them; 784 m multiplies the area by 4 without taking the square root first.
  8. c) It is not square: a right angle would make that distance 100 cm. — For a right angle, the distance d must satisfy d² = 60² + 80² = 3600 + 6400 = 10 000, so d = 100 cm — the 3-4-5 triangle made 20 times bigger. 97 cm is too short, so the corner is a little less than 90°. 140 cm adds the sides instead of their squares.
  9. c) 40 m — Walking along two sides is 80 + 60 = 140 m. The diagonal is the hypotenuse, √(80² + 60²) = √10 000 = 100 m, so she saves 140 − 100 = 40 m. 100 m is the length of the shortcut, not how much shorter it is.
  10. d) 6, √40, 7, √50 — √40 is between 6 and 7, because 6² = 36 and 7² = 49, and √50 is just over 7, because 50 is just over 49. Putting √50 before 7 misses that 50 is more than 7² = 49, and putting both roots last compares 40 and 50 with 6 and 7 as if the root signs were not there.