The Pythagorean Theorem · Grade 8

How 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.

Learning objectives

The triangle hidden in the grid

Two points on a plane define exactly one right triangle: the segment joining them is the hypotenuse, and the horizontal and vertical gaps are the legs. The right angle sits at the point directly below one and level with the other.

So there is no new formula here. Subtract the x-coordinates, subtract the y-coordinates, and apply the theorem to those two differences.

The sign of each difference does not matter, because both are squared. Subtract in either order and take the size; the result is the same. A distance cannot come out negative.

Diagonals and heights

A diagonal cuts a rectangle into two right triangles whose legs are its length and width. The distance between two opposite corners is exactly their hypotenuse.

The height of an isosceles triangle drops to the midpoint of the base and makes a right angle there. The legs are the height and half the base, and the equal side is the hypotenuse — the same calculation once again.

Word problems

A ladder against a wall, a boat's mast and its stay, the distance between two towns on a map: the first step is always the same, which is to draw it and mark where the right angle is. Most of the difficulty in these questions is the translation into a triangle, not the arithmetic.

At the end, check the answer makes sense in the situation described. A ladder reaching higher than its own length, or a distance greater than the sum of the two legs, signals an error — the hypotenuse is always shorter than the legs added together and longer than either one.

Worked examples

  1. What is the distance between the points (1,2) and (4,6)?

    1. Horizontal gap: 4 minus 1 is 3
    2. Vertical gap: 6 minus 2 is 4
    3. Apply the theorem: 9 plus 16 is 25
    4. Take the root: 5

    Answer: The distance is 5

  2. A rectangle is 12 long and 5 wide. How long is its diagonal?

    1. The diagonal is the hypotenuse; length and width are the legs
    2. Square them: 144 and 25
    3. Add: 169
    4. Take the root: 13

    Answer: The diagonal is 13

  3. An isosceles triangle has a base of 10 and equal sides of 13. What is the height to the base?

    1. The height falls to the midpoint, giving a leg of 5
    2. The equal side is the hypotenuse, so subtract: 169 minus 25 is 144
    3. Take the root: 12

    Answer: The height is 12

Common mistakes

Adding the coordinates instead of subtracting them
A leg is the difference between coordinates, not their sum. Adding gives a much larger distance, and it also changes if you slide both points together — which a distance does not.
Using the equal side as a leg in an isosceles triangle
The equal side is the hypotenuse of the half-triangle; the legs are the height and half the base. Swapping them produces a height longer than the equal side, which is impossible.
Using the whole base instead of half of it
The height of an isosceles triangle meets the base exactly at its midpoint. Using the full base doubles that leg and wrecks the entire calculation.

What to remember

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