Pythagorean Theorem
In short: Pythagorean theorem: for any right triangle, squaring the two shorter sides and adding gives the square of the longest side, a² + b² = c².
8th GradeGeometry8.G.B.7
What is Pythagorean Theorem?
Build a square on every side of a right triangle and the two smaller squares will together cover the same area as the largest one, a surprising fact called the Pythagorean theorem that requires a \(90^{\circ}\) corner.
Call the short sides \(a\) and \(b\) and the long side \(c\); then \(a^2 + b^2 = c^2\). You can recover a missing side from the other two, and the converse tests whether three given lengths form a right triangle.
How to spot it
- A diagram has a right-angle box and exactly one side length is missing.
- The story features a ladder on a wall, a ramp, a screen diagonal or a rectangle’s diagonal.
- A question asks for the shortest route across a rectangular field or room.
- Three side lengths are listed and you are asked if the triangle is right.
The method
- Mark the right angle; the side facing it is the hypotenuse \(c\), and the other two are \(a\) and \(b\).
- Write the Pythagorean theorem, \(a^2 + b^2 = c^2\), and substitute the two known lengths.
- Square the numbers, then add them when seeking \(c\) or subtract when seeking a leg.
- Take the positive square root and attach the unit.
Pythagorean Theorem: solved example
Problem
A 6.5-meter ladder rests against a barn wall with its top touching the wall 6 meters above the ground. How far from the wall is the foot of the ladder?
Solution
The ladder faces the right angle, so \(c = 6.5\), and one leg is the wall height, 6. Let \(x\) be the unknown distance: \(x^2 + 6^2 = 6.5^2\).
Then \(x^2 + 36 = 42.25\), so \(x^2 = 6.25\) and \(x = 2.5\). The foot of the ladder sits 2.5 meters from the wall.
Try it
Question 1
A right triangle has legs of 7 in. and 24 in. How long is the hypotenuse?
Show the answer
\(49 + 576 = 625\), and \(\sqrt{625} = 25\), so 25 in.
Question 2
The hypotenuse measures 10 and one leg measures 7. Find the other leg to the nearest hundredth.
Show the answer
\(100 – 49 = 51\), and \(\sqrt{51} \approx 7.14\).
Pythagorean Theorem vs. Distance Formula
The distance formula is this rule dressed for a coordinate grid: the horizontal and vertical gaps between two points are the legs, and the distance is the long side. The theorem handles any right triangle anywhere; the formula starts directly from coordinates.
Keep going
Lessons
Worksheets and quizzes
- The Pythagorean Theorem – Worksheet
- The Pythagorean Theorem – Quiz
- Distance on the Coordinate Plane – Worksheet
- Distance on the Coordinate Plane – Quiz
Related words
Whole dictionary: A–Z · 8th Grade · Geometry