Grades Grade 8 GeometryThe Pythagorean Theorem

The Pythagorean Theorem

📐 Geometry

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The Pythagorean Theorem

In any right triangle, the square of the hypotenuse equals the sum of the squares of the legs.

The Pythagorean Theorem applies to right triangles. The two shorter sides (legs) are a and b; the longest side (hypotenuse, opposite the right angle) is c. The theorem and its converse let you find missing side lengths and verify whether a triangle is a right triangle.

a^2 + b^2 = c^2

🏗️Finding the hypotenuse

A right triangle has legs of 6 cm and 8 cm. Find the hypotenuse.

6² + 8² = c²
36 + 64 = c²
100 = c²
c = √100 = 10 cm

6^2 + 8^2 = c^2 \Rightarrow 36 + 64 = 100 \Rightarrow c = \sqrt{100} = 10

📍Distance between two points

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

Δx = 5 − 1 = 4, Δy = 5 − 2 = 3
d = √(4² + 3²) = √(16 + 9) = √25 = 5

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Using the converse to classify a triangle:

1Identify the longest side — this is the candidate for c

2Compute a² + b² using the two shorter sides

3If a² + b² = c², the triangle is a right triangle

4If a² + b² > c², it is acute; if a² + b² < c², it is obtuse

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Remember This!

Memorize common Pythagorean triples: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25). Their multiples also work — (6, 8, 10) and (9, 12, 15) are valid right triangles.

✏️ Try It!

A right triangle has a hypotenuse of 13 and one leg of 5. What is the length of the other leg?

Practice with CalcVerse

🔺Pythagorean Theorem

Take Quiz 📝 — 25 Questions