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What Is the Pythagorean Theorem?(With Examples)

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the lengths of the other two sides.

Grade 8KS3 / GCSE FoundationÅrskurs 8-9

📖Definition

📐Formula

a² + b² = c²

Where a and b are the lengths of the two legs (shorter sides), and c is the length of the hypotenuse (the longest side, opposite the right angle).

📝Step-by-Step Guide

Identify the sides

Determine which side is the hypotenuse (opposite the right angle) and which are the legs.

Set up the equation

Use a² + b² = c² with the known values.

a² + b² = c²

Solve for the unknown

If finding the hypotenuse: c = √(a² + b²). If finding a leg: a = √(c² - b²).

Calculate and check

Compute the final value and verify your answer makes sense geometrically.

⚠️Common Mistakes to Avoid

Confusing the hypotenuse with a leg
Forgetting to take the square root at the end
Using the formula for non-right triangles
Calculation errors when squaring numbers

✏️Practice Problems

Easy

Find the hypotenuse when a = 3 and b = 4

Answer: c = 5

Medium

Find side b when a = 5 and c = 13

Answer: b = 12

Hard

A ladder is 10m long and leans against a wall, reaching 8m high. How far from the wall is the base?

Answer: 6 meters

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Worked Examples

Find the Hypotenuse (3-4-5 Triangle)easy

Pythagorean Theorem Examplemedium

Right Triangles Distance Formula Introduction to Trigonometry

Curriculum Alignment

CommonCore (8.G.B.7)GCSE

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