Grade 8

The Pythagorean Theorem

Right triangles from square roots through the theorem and its converse, up to distances between points in the coordinate plane.

6 lessons

  1. 1What is a square number and what is a square root?Squaring a number means multiplying it by itself, and taking a square root is the opposite operation — it gives back the number the square came from. Seven squared is forty-nine, so the square root of forty-nine is seven. These two operations are everything Pythagoras' theorem is made of.30m
  2. 2What are the legs and the hypotenuse of a right triangle?The hypotenuse is the side opposite the right angle, and it is always the longest side of the triangle. The other two sides are called legs, and they are the ones that form the right angle between them. Every use of Pythagoras' theorem begins by identifying these three correctly, not by calculating.25m
  3. 3What does Pythagoras' theorem say and why is it true?The theorem says that the sum of the squares of the two legs equals the square of the hypotenuse. Put another way, the area of the square built on the hypotenuse equals the combined area of the squares built on the legs. It holds for every right triangle without exception, and you can see why using areas.40m
  4. 4How do you find a leg when the hypotenuse is known?The same theorem, run backwards: you subtract instead of adding. The missing leg squared equals the hypotenuse squared minus the known leg squared, and then you take the root. The step that decides everything is checking first which of the given sides really is the hypotenuse, because subtracting the wrong way round gives a negative number.35m
  5. 5How do you check whether a triangle is right-angled?Test the three sides using the converse: if the squares of the two shorter sides add up to exactly the square of the longest, the triangle is right-angled, and the right angle sits opposite that longest side. If the sum is larger, that angle is acute; if smaller, it is obtuse.30m
  6. 6How do you find the distance between two points and a diagonal in a rectangle?Subtract the coordinates from one another to get the two legs — the horizontal gap and the vertical gap — then apply Pythagoras' theorem to them. A diagonal in a rectangle is exactly the same calculation, with the length and width as the legs. Every problem of this kind is a right triangle in disguise.40m