Pythagoras’ Theorem

Pythagoras’ Theorem tells us that the sides of a right angled triangle are related in the following way:

    \[(\text{shorter}_1)^2+(\text{shorter}_2)^2=(\text{hypotenuse})^2\]

If we begin by labelling our triangle with letters a, b and c as on the following triangle,

we have

    \[a^2+b^2=c^2\]

This is a beautifully simple theorem, however it is not that obvious why it works. It is named after Pythagoras, not because he discovered it but because he was probably the first to explain why it work – or the first to have his explanation recorded.

Short story part 1 and short story part 2 offer an explanation of why this theorem works.


Visual explanation of the Pythagorean Theorem

The diagram on the left is constructed with four congruent triangles (four ‘equal’ triangles) and one square.

The diagram on the right is constructed with the same four congruent triangles and with two smaller squares to fill in the gaps.

Made by Daniel Pearcy.

Without using any algebra, this illustrates why c^2 must be equal to a^2+b^2.

Notice that the four triangles in the first diagram have a match on the second diagram. That means that the area of the center green square, c\times c on the left diagram must equal the areas of the small dark green square, a\times a and small blue square, b\times b together on the right diagram. That is, c^2=a^2+b^2.

 


Sometimes the letters in a formula and the letters in a problem might not match up. For example, the formula c^2=a^2+b^2 assumes that we label the hypotenuse c. I practice the hypotenuse might be called x or y or b or fence or anything else for that matter. That is why it is important to remember that it is the shorter sides squared, then added, that give the hypotenuse squared.

Example 1 The longest side (opposite the right angle) is not known

Intuitively: To find the longest length, add the two shorter lengths (squared).

    \begin{align*}9.1^2+4.1^2&=x^2\\[10pt] 99.62&=x^2 \\[10pt]x&=\sqrt{99.62}\\[10pt] x&=9.98 \text{ (2 d.p.)}\end{align*}

Example 2 A shorter side is not known

Intuitively: To find a shorter length, start with the longest length (squared) and take the other shorter length (squared).

    \begin{align*}7.8^2-6.8^2&=x^2\\[10pt]14.6&=x^2\\[10pt]x&=\sqrt{14.6}\\[10pt]x&=3.82 \text{ (2 d.p.)}\end{align*}

 

Calculating the third side on a Right Angled Triangle


All of the above from Mathisfun and Khan Academy


Top BC G10 Menu Geometry and Trigonometry