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Step 10: Pre-algebra

Pythagorean theorem

Step 10 · Pre-algebra topics

Use a² + b² = c² to test for right triangles, find a missing side of a right triangle, and find the distance between two points on the coordinate plane.

The theorem

In a right triangle, the longest side, opposite the right angle, is the hypotenuse, c. The other two sides are the legs, a and b. The Pythagorean theorem says:

a2+b2=c2a^{2} + b^{2} = c^{2}
Squares on the sides of a right triangle with sides 3, 4 and 5: 9 + 16 = 25.

Example: Checking for a right angle

Is a triangle with sides 6, 8 and 10 a right triangle?

  1. 6² + 8² = 36 + 64 = 100.
  2. 10² = 100 as well.

AnswerYes

Finding a missing side

Use the theorem to find a missing side. For the hypotenuse, add the squares of the legs; for a leg, subtract.

Example: The hypotenuse

The legs are 5 and 12. How long is the hypotenuse?

  1. 5² + 12² = 25 + 144 = 169.
  2. c = √169 = 13.

Answer13

Example: A leg

The hypotenuse is 17 and one leg is 8. How long is the other leg?

  1. 17² − 8² = 289 − 64 = 225.
  2. b = √225 = 15.

Answer15

Example: A ladder

A 10 ft ladder leans on a wall with its foot 6 ft from the wall. How high up the wall does it reach?

  1. 10² − 6² = 100 − 36 = 64.
  2. √64 = 8.

Answer8 ft

Distance on the coordinate plane

To find the distance between two points, draw a right triangle: the across gap and the up gap are the legs, and the distance is the hypotenuse.

From (1, 1) to (7, 5): across 6 and up 4, so the distance is √(36 + 16) = √52, about 7.2.

Example: Distance

How far is it from (−2, 1) to (4, 9)?

  1. Across: 4 − (−2) is 6. Up: 9 − 1 = 8.
  2. 6² + 8² = 36 + 64 = 100.
  3. √100 = 10.

Answer10 units

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2}}

Try it yourself

Work each one on paper first, then open the answer to check.

  1. The legs of a right triangle are 9 and 12. How long is the hypotenuse?

    Show answer

    15

  2. The hypotenuse is 25 and one leg is 7. How long is the other leg?

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    24

  3. Is a triangle with sides 5, 6 and 8 a right triangle?

    Show answer

    No: 25 + 36 = 61, but 8² = 64.

  4. A rectangle is 8 cm by 15 cm. How long is its diagonal?

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    17 cm

  5. How far is it from (0, 0) to (5, 12)?

    Show answer

    13 units

  6. How far is it from (1, 2) to (4, 6)?

    Show answer

    5 units

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