The converse of the Pythagorean theorem; applications and distances

About this lesson

Grade 8 › Pythagorean theorem and irrational numbers

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The converse of the Pythagorean theorem; applications and distances

Warm-up

Three quick questions from earlier lessons.

Discover

A triangle has sides 6, 8 and 10. Is it a right triangle?

  1. Yes
  2. You cannot tell

Key idea

The converse, and right triangles everywhere.

Converse: if a2+b2=c2a^2 + b^2 = c^2 for the two shorter sides aa, bb and the longest side cc, the triangle is a right triangle. Use the theorem for ladders, diagonals and distances between points: the horizontal and vertical changes are the legs.

Worked example

Distance from (1,2)(1, 2) to (7,10)(7, 10).

Legs 6 and 8
Changes.
Distance 10
√(36 + 64).

Your turn, guided

Your turn: distance from (0,0)(0, 0) to (5,12)(5, 12).

1313
√(25 + 144).

Practice ★

  1. Is a triangle with sides 3, 4 and 5 a right triangle?
    1. Yes: the squares of the two shorter sides add up to the square of the longest
    2. No: the sides are not consecutive whole numbers
    3. You cannot tell without measuring an angle
  2. Is a triangle with sides 9, 12 and 15 a right triangle?
    1. Yes: the squares of the two shorter sides add up to the square of the longest
    2. No: the sides are not consecutive whole numbers
    3. You cannot tell without measuring an angle
  1. A 13 ft ladder leans against a wall and reaches 12 ft up. How far is its foot from the wall?

Watch out!

Practice ★★

  1. Is a triangle with sides 7, 8 and 9 a right triangle?
    1. No: 49 + 64 is not 81
    2. Yes: like 3, 4, 5, the sides are consecutive
    3. You cannot tell without measuring an angle
  2. Is a triangle with sides 6, 7 and 8 a right triangle?
    1. No: 36 + 49 is not 64
    2. Yes: like 3, 4, 5, the sides are consecutive
    3. You cannot tell without measuring an angle
  1. Boss problem. Find the distance between A(−1,−5)A(-1, -5) and B(5,3)B(5, 3) using a right triangle.

Exit ticket

Three questions to check you have got it.

  1. A 13 ft ladder leans against a wall and reaches 12 ft up. How far is its foot from the wall?
  2. A 39 ft ladder leans against a wall and reaches 36 ft up. How far is its foot from the wall?
  3. A 26 ft ladder leans against a wall and reaches 24 ft up. How far is its foot from the wall?

Summary card

Converse and applications

a2+b2=c2a^2 + b^2 = c^2 means right angle; distances are hypotenuses.

Watch out: always test with the longest side as cc.

End of the lesson

Next lesson: Distance and midpoint in the coordinate plane Back to the chapter

Answers: The converse of the Pythagorean theorem; applications and distances

A

  1. Yes: the squares of the two shorter sides add up to the square of the longest
  2. Yes: the squares of the two shorter sides add up to the square of the longest

B

  1. 55

C

  1. No: 49 + 64 is not 81
  2. No: 36 + 49 is not 64

D

  1. 1010

E

  1. 55
  2. 1515
  3. 1010

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