Sec 2 Pythagoras’ Theorem | Right Triangles, Hypotenuse and Distance

Pythagoras’ Theorem connects the three side lengths of a right-angled triangle. The theorem is simple: a²+b²=c², where c is the hypotenuse, the side opposite the right angle. The difficulty is usually not the formula; it is identifying the right triangle, the hypotenuse and whether the unknown side should be found by addition or subtraction.

Students searching for Pythagoras theorem Sec 2, right triangle, hypotenuse or Secondary 2 geometry often substitute numbers before deciding which side is c. The reliable method is mark the right angle → identify the hypotenuse → label the sides → write the equation → solve → check against geometry.

This page is the canonical Pythagoras owner under the Secondary Mathematics Topic Library.

Quick answer: the theorem

a²+b²=c²

c is always the hypotenuse, the longest side and the side opposite the 90° angle.

Worked example 1: find hypotenuse

Legs: 6 cm and 8 cm.

c² = 6²+8² = 36+64 = 100.

c = 10 cm.

Worked example 2: find a shorter side

Hypotenuse: 13 cm. One leg: 5 cm.

Other leg² = 13²−5² = 169−25 = 144.

Other leg = 12 cm.

Subtraction is used because the hypotenuse square is the total of the two leg squares.

The hypotenuse check

opposite the right angle
longest side
never one of the two perpendicular sides

If the answer for the hypotenuse is smaller than one leg, something is wrong.

Square root at the end

Pythagoras first gives a squared length. If x²=49, then a geometric side length is x=7, not 49.

Use the positive root because physical length is positive.

Worked example 3: coordinate-style distance

A horizontal separation is 9 units and vertical separation is 12 units.

Direct distance = √(9²+12²)=√225=15 units.

This anticipates later coordinate geometry.

Pythagorean triples

Some integer side sets appear often:

3,4,5
5,12,13
8,15,17
7,24,25

Recognising them can speed checking, but the theorem should still be understood rather than memorised as a list.

Converse of Pythagoras

If three side lengths satisfy a²+b²=c² with c the largest side, the triangle is right-angled.

Example: 9,12,15: 9²+12²=81+144=225=15², so the triangle is right-angled.

Worked example 4: test a triangle

Sides 6,8,11.

6²+8²=100, but 11²=121.

The triangle is not right-angled.

Pythagoras inside composite figures

Many questions hide a right triangle inside a rectangle, prism, ladder problem or coordinate diagram. Draw or highlight the relevant triangle first.

Do not apply the theorem to arbitrary triangles.

Ladder problem

A ladder 10 m long leans against a wall. Its foot is 6 m from the wall.

Height² = 10²−6² = 100−36 = 64.

Height = 8 m.

Exact versus rounded answers

If the square root is not exact, keep enough calculator precision and round only as required by the question.

Avoid rounding intermediate values too early.

Common errors

chooses wrong side as hypotenuse
adds squares when finding a shorter side
forgets square root
applies theorem to a non-right triangle
rounds too early
uses unsquared side lengths in the equation

The Pythagoras audit

Where is the right angle?
Which side is opposite it?
Am I finding the hypotenuse or a leg?
Should I add or subtract squared lengths?
Did I take the square root?
Is the final length geometrically sensible?

Practice

Practice 1

Question: legs 9,12

Answer: hypotenuse 15

Practice 2

Question: hypotenuse 17, leg 8

Answer: other leg 15

Practice 3

Question: sides 5,12,13

Answer: right-angled

Practice 4

Question: sides 7,8,10

Answer: not right-angled

Practice 5

Question: ladder 13, base 5

Answer: height 12

Frequently asked questions

How do I identify the hypotenuse?

It is opposite the right angle and is the longest side.

When do I subtract?

When the hypotenuse and one leg are known and you are finding the other leg.

Can I use Pythagoras on any triangle?

No. The standard theorem applies to right-angled triangles.

Why do I take a square root?

Because the theorem relates squared lengths; solving for the actual side length requires reversing the square.

Where does this sit in Atlas?

This is the canonical Sec 2 Pythagoras owner under Secondary Mathematics Topic Library.

The final Pythagoras rule

Find the right angle before finding the formula. Once the hypotenuse is identified correctly, the theorem becomes a reliable relationship rather than a memory trick.

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