Pythagoras and Coordinate Distances

Identify the hypotenuse

The hypotenuse is opposite the right angle and is the longest side. Call it c. The two shorter sides are a and b. For a right-angled triangle, a²+b²=c². The letters may change in a question; the position opposite 90° decides which side is the hypotenuse.

Finding the longest side

Square both shorter lengths, add the squares, then take the positive square root. The sum is c², not c.

Worked example — two shorter sides

A right triangle has shorter sides 6 cm and 8 cm. c²=6²+8²=36+64=100. Therefore c=√100=10 cm. The answer is longer than either given side.

Worked example — an exact answer

With shorter sides 3 cm and 5 cm, c=√(9+25)=√34 cm. This is exact. If asked for one decimal place, c≈5.8 cm. Keep the square root on your calculator until the final step.

Finding a shorter side

When the hypotenuse is known, subtract the square of the known shorter side from the square of the hypotenuse. Rearrange first: a²=c²−b².

Worked example — a ladder

A 13 m ladder reaches 12 m vertically up a wall on level ground. The wall and ground meet at 90°. The ladder is the hypotenuse. If the foot is x metres from the wall, x²+12²=13², so x²=169−144=25 and x=5 m. Adding here would give a distance longer than the ladder, which is impossible.

Coordinate distances

Join two coordinate points and draw horizontal and vertical lines to make a right triangle. Subtract x-coordinates for its horizontal length and y-coordinates for its vertical length. Use positive lengths; squaring removes a negative difference anyway.

Worked example — across quadrants

A(−2,1) and B(4,9) differ by 4−(−2)=6 horizontally and 9−1=8 vertically. AB=√(6²+8²)=10 units. The midpoint is a different question: averaging coordinates gives (1,5).

In general, distance=√[(x₂−x₁)²+(y₂−y₁)²]. For a horizontal or vertical segment, one difference is zero.

Testing for a right angle and explaining the theorem

To test three side lengths, first identify the longest. A triangle with sides 5,12,13 is right-angled because 5²+12²=13². Sides 5,6,8 do not give a right angle: 25+36≠64. Do not use the rule on a triangle just because it looks right-angled.

Why a²+b²=c²

Arrange four identical right triangles, with shorter sides a and b, inside a square of side a+b. Their hypotenuses bound a central square of side c: adjacent acute angles sum to 90°, so its corners are right angles. The large square has area (a+b)². The four triangles have total area 4×½ab=2ab. Therefore c²=(a+b)²−2ab=a²+b².

Think Like a Mathematician

Which side must be longest before you calculate? That prediction tells you whether to add or subtract and provides a quick check afterwards.

Common Mistakes

Do not square a sum: 6²+8² is 100, whereas (6+8)² is 196. Do not stop at c². Use consistent units and give a length answer, not square units. Establish a right angle before using the theorem.

Practice — Pythagoras and distances

1. A right triangle has shorter sides 9 cm and 12 cm. Find its hypotenuse.

2. A right triangle has hypotenuse 17 m and one shorter side 8 m. Find the other.

3. Find the exact hypotenuse when the shorter sides are 2 cm and 5 cm.

4. Find the distance from (−3,−1) to (5,5).

5. Are the sides 7,24,25 those of a right triangle? Explain.

6. A rectangle is 1.2 m by 50 cm. Find its diagonal in metres.

Frequently Asked Questions

Can I use this for every triangle?

No. The theorem requires a right angle. Non-right-angled triangles need another method, such as the cosine rule at Higher.

Why take only the positive square root?

A geometric side length is positive. The negative algebraic root does not represent a length.

Continue Learning

Previous: Similar Shapes: Length, Area and Volume

Next: Right-Angled Trigonometry and Exact Values

Geometry and Measures topic overview