Grasp Maths

Year 9

Pythagoras' Theorem in 2D

Apply Pythagoras' theorem a2+b2=c2a^2 + b^2 = c^2 to find any missing side in a right-angled triangle, and determine whether a triangle is right-angled.

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Lesson overview

Geometry — right-angled triangles

Pythagoras' theorem states that in any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a2+b2=c2a^2 + b^2 = c^2 where cc is the hypotenuse — the longest side, always opposite the right angle. To find the hypotenuse: c=a2+b2c = \sqrt{a^2 + b^2}. To find a shorter side: a=c2−b2a = \sqrt{c^2 - b^2}.

Finding the hypotenuse

A right-angled triangle has legs a=3a = 3 cm and b=4b = 4 cm. Find cc.

a = 3 cmb = 4 cmc = ?a² + b² = c²

The 3–4–5 is a Pythagorean triple. Always identify cc (hypotenuse) before substituting.

Finding a shorter side — rearrange first

The hypotenuse is c=13c = 13 cm and one leg is b=5b = 5 cm. Find leg aa.

When finding a shorter side, subtract: a=c2−b2a = \sqrt{c^2 - b^2}. Always identify the hypotenuse first.

Non-integer answers — leave as surd or round sensibly

A right-angled triangle has legs 55 cm and 77 cm. Find the hypotenuse.

74\sqrt{74} is irrational. Unless told otherwise, round to 3 significant figures or 2 decimal places.

Testing whether a triangle is right-angled

Is a triangle with sides 88 cm, 1515 cm and 1717 cm right-angled?

VerdictYes — right-angled

because 82+152=1728^2 + 15^2 = 17^2

Test: does a2+b2=c2a^2 + b^2 = c^2 hold? Use the longest side as cc.

Worked example — ladder against a wall

A ladder of length 1010 m leans against a wall. The foot of the ladder is 44 m from the wall. How high up the wall does it reach? Give your answer to 2 decimal places.

h = ?4 m10 m (ladder)h² = c² − b²
  1. Identify the hypotenuse: the ladder (1010 m) is the longest side and opposite the right angle.
  2. The base (44 m) and the height (hh) are the two legs.
  3. Rearrange: h2=c2−b2=102−42=100−16=84h^2 = c^2 - b^2 = 10^2 - 4^2 = 100 - 16 = 84.
  4. Take the positive square root: h=84≈9.17h = \sqrt{84} \approx 9.17 m.
  5. Always check: 42+9.172≈16+84=100=1024^2 + 9.17^2 \approx 16 + 84 = 100 = 10^2 ✓

Try it

Apply $a^2 + b^2 = c^2$ to each question. Show your working where possible.

Question 1

A right-angled triangle has legs 66 cm and 88 cm. Find the hypotenuse.

💡 Use c2=a2+b2=62+82c^2 = a^2 + b^2 = 6^2 + 8^2.

Question 2

The hypotenuse is 2525 cm and one leg is 77 cm. Find the other leg.

💡 Rearrange: a2=c2−b2=252−72a^2 = c^2 - b^2 = 25^2 - 7^2.

Question 3

A right-angled triangle has legs 55 cm and 1212 cm. What is the hypotenuse?

💡 52+122=25+144=169=1325^2 + 12^2 = 25 + 144 = 169 = 13^2.

Question 4

Is a triangle with sides 99 cm, 1212 cm and 1515 cm right-angled?

💡 Test: 92+122=?9^2 + 12^2 = ? and 152=?15^2 = ?

Question 5

A right-angled triangle has legs 77 cm and 99 cm. Which is the hypotenuse to 2 d.p.?

💡 c=72+92=49+81=130c = \sqrt{7^2 + 9^2} = \sqrt{49 + 81} = \sqrt{130}

Common mistakes

Watch for these when working through the lesson.

  • Forgetting that cc must be the hypotenuse — using a leg as cc gives a wrong answer.
  • Subtracting when finding the hypotenuse: you always add a2+b2a^2 + b^2.
  • Not square-rooting at the end: c2=100c^2 = 100 does not mean c=100c = 100.
  • Rounding too early — keep the full calculator value until the final answer.

Related topics

These ideas fit closely with this lesson.

  • Surds and irrational numbers
  • Trigonometry in right-angled triangles (sin, cos, tan)
  • Pythagoras in 3D
  • Circle theorems — angle in a semicircle

Practice next

Independent practice will plug in here

This lesson builds the understanding first. Deeper adaptive practice can sit here later.