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Pythagorean Relationship

3.0 Measurements · 3.1 Pythagorean Relationship

Syllabus tag: Kenya CBC | Grade 7 Mathematics | Strand 3.0 Measurements | Sub-Strand 3.1 Pythagorean Relationship (4 lessons)

Lesson objectives

By the end of this topic, you will be able to:

  • Recognise the relationship of the sides of a right-angled triangle
  • Identify Pythagorean relationships in different situations
  • Apply the Pythagorean relationship to real-life situations
  • Promote the use of Pythagoras' Theorem in real-life situations

Pythagorean Relationship

Take any right-angled triangle. Square the two short sides, add them, and you get the square of the long side. Every time, for every such triangle.

a) The sides

A right-angled triangle A right-angled triangle 8 cm 6 cm c

The hypotenuse is the longest side. It is always opposite the right angle, never one of the sides forming it.

Identify the hypotenuse before doing anything else. Getting it wrong turns an addition into a subtraction.

b) The relationship

Take a right-angled triangle with short sides a and b, and hypotenuse c. Then:

a² + b² = c²

This only works for right-angled triangles. A triangle with no right angle does not obey it.

c) Finding the hypotenuse

Finding the hypotenuse Finding the hypotenuse rule → a² + b² = c² the two short sides and the long one c² = 6² + 8² square each short side c² = 36 + 64 work them out c² = 100 add c = 10 cm take the square root

Square both short sides, add them, then take the square root.

The 3, 4, 5 triangle is the smallest whole-number example, since 9 + 16 = 25. Doubling it gives 6, 8, 10, which is the triangle above.

Other useful sets are 5, 12, 13 and 8, 15, 17. These are called Pythagorean triples.

d) Finding a short side

If you know the hypotenuse and one short side, subtract instead of adding.

Finding a short side Finding a short side rule → a² + b² = c² the same rule 5² + b² = 13² hypotenuse is 13 b² = 169 − 25 subtract this time b² = 144 work it out b = 12 cm take the square root

Take care which way round. The hypotenuse squared is always the larger, so it comes first in the subtraction.

A short side longer than the hypotenuse is impossible. If that happens, you added when you should have subtracted.

e) Checking whether an angle is right

The relationship works backwards too. If a² + b² equals c², the triangle has a right angle. If it does not, the triangle does not.

A builder uses this to check a corner. Measure 3 m along one wall and 4 m along the other. If the diagonal is exactly 5 m, the corner is square.

f) Where this is used

A carpenter checks a frame is square. A builder sets out a foundation. Finding the length of a ladder against a wall uses it too.

Words to know

  • Right-angled triangle — a triangle with one angle of exactly 90°.
  • Hypotenuse — the longest side of a right-angled triangle, opposite the right angle.
  • Leg (of a right-angled triangle) — either of the two shorter sides forming the right angle.
  • Pythagorean relationship — the rule that a² + b² = c² for the legs a, b and hypotenuse c of a right-angled triangle.

:::checkpoint Check yourself

  1. Which side is the hypotenuse?
  2. A triangle has short sides 9 cm and 12 cm. Find the hypotenuse.
  3. A triangle has hypotenuse 25 cm and one short side 7 cm. Find the other.
  4. Do sides 5, 6 and 8 make a right-angled triangle? :::

Bridge to practice

Try the exercises below — on identifying and applying the Pythagorean relationship.

Check yourselfPractise Pythagorean Relationship — 10 questions →Next in MathematicsWhole Numbers →