Learning Objectives
  • State and apply Pythagoras’ theorem in right-angled triangles.
  • Identify the hypotenuse correctly.
  • Find unknown sides and solve two-dimensional problems.
  • Use the converse to test whether a triangle is right-angled.
  • Apply Pythagoras to coordinates and compound diagrams.
Key Terms
Right-angled triangle
A triangle containing one 90^\circ angle.
Hypotenuse
The side opposite the right angle and the longest side.
Leg
Either of the two sides forming the right angle.
Converse
The reverse statement used to test whether a triangle is right-angled.
Exact form
An unsimplified or simplified surd answer such as 5\sqrt2.
8th Edition Chapter Map
  • The theorem and hypotenuse identification
  • Finding hypotenuse and shorter sides
  • Converse of Pythagoras
  • Coordinate and diagonal problems
  • Compound two-dimensional applications
The Theorem

For a right-angled triangle with legs a,b and hypotenuse c, the square of the hypotenuse equals the sum of the squares of the other two sides.

a^2+b^2=c^2

The hypotenuse is always opposite the right angle. It is not automatically the side drawn sloping or at the top of a diagram.

Worked Example: Find The Hypotenuse

Question: A right triangle has legs 9 cm and 12 cm. Find the hypotenuse.

  1. c^2=9^2+12^2=81+144=225.
  2. c=\sqrt{225}=15.

Answer: 15 cm.

Finding A Shorter Side

When the hypotenuse is known, subtract the square of the known leg from the square of the hypotenuse.

Worked Example: Find A Leg

Question: The hypotenuse is 17 m and one leg is 8 m. Find the other leg.

  1. x^2=17^2-8^2=289-64=225.
  2. x=15.

Answer: 15 m.

Exact And Rounded Answers

If the square root is not exact, leave an exact surd when requested or use a calculator and round according to the question. For example, \sqrt{50}=5\sqrt2\approx7.071. Do not round intermediate squared values unnecessarily.

The Converse

If the longest side is c and a^2+b^2=c^2, the triangle is right-angled. If the equality does not hold, it is not right-angled.

Worked Example: Test A Triangle

Question: Determine whether sides 7, 24 and 25 form a right triangle.

  1. The longest side is 25.
  2. 7^2+24^2=49+576=625.
  3. 25^2=625.

Answer: Yes, the triangle is right-angled.

Diagonals Of Rectangles And Squares

A rectangle’s diagonal forms the hypotenuse of a right triangle whose legs are the length and width. A square of side s has diagonal s\sqrt2.

Worked Example: Rectangle Diagonal

Question: Find the diagonal of a rectangle 14 cm by 9 cm.

  1. d^2=14^2+9^2=277.
  2. d=\sqrt{277}\approx16.643.

Answer: 16.6 cm to 3 significant figures.

Distance Between Two Coordinate Points

The horizontal and vertical differences form the legs of a right triangle. Thus the distance between (x_1,y_1) and (x_2,y_2) follows directly from Pythagoras.

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
Worked Example: Coordinate Distance

Question: Find the distance between (-2,3) and (4,-5).

  1. Horizontal difference =4-(-2)=6.
  2. Vertical difference =-5-3=-8.
  3. d=\sqrt{6^2+(-8)^2}=\sqrt{100}=10.

Answer: 10 units.

Compound Diagrams

Some problems require Pythagoras more than once. Mark an intermediate diagonal, calculate it, then use it in a second right triangle. Keep the exact value as long as possible.

Scope Note

This chapter develops two-dimensional use of Pythagoras. Three-dimensional applications are treated with later trigonometry and mensuration topics.

Examination Guidance
  • Mark the right angle and identify the hypotenuse before writing an equation.
  • Write the theorem with the required unknown as the subject.
  • Use exact values until the final rounding step.
  • Include units and state the requested accuracy.
Common Mistakes
  • Adding the sides instead of their squares.
  • Subtracting in the wrong order and obtaining a negative square.
  • Using a non-hypotenuse side as c.
  • Forgetting the final square root.
Knowledge Check And Practice

1. Find the hypotenuse when the legs are 5 cm and 12 cm.

Answer: 13 cm.

2. Find the shorter side when the hypotenuse is 10 m and the other leg is 6 m.

Answer: 8 m.

3. Do sides 9, 40 and 41 form a right triangle?

Answer: Yes, because 9^2+40^2=41^2.

4. Find the diagonal of a square of side 7 cm.

Answer: 7\sqrt2 cm, approximately 9.90 cm.

5. Find the distance between (1,2) and (7,10).

Answer: \sqrt{6^2+8^2}=10.
Extended Worked Practice
Diagonal Of A Rectangle

Question: A rectangle is 9 cm by 40 cm. Find its diagonal.

d^2=9^2+40^2=81+1600=1681, so d=41 cm.

Side Of A Rhombus From Its Diagonals

Question: The diagonals of a rhombus are 16 cm and 30 cm. Find its side length.

  1. The diagonals bisect each other at right angles, giving half-lengths 8 and 15.
  2. s^2=8^2+15^2=64+225=289.

Answer: s=17 cm.

Exact Surd Length

Question: A right triangle has legs 5 cm and 7 cm. Give the hypotenuse exactly and to 3 significant figures.

c=\sqrt{5^2+7^2}=\sqrt{74} cm \approx8.60 cm.

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