The Pythagorean Theorem: Practice

The hypotenuse is the side across from the right angle, so it is the one side that does not touch $B$.

$\angle ABC = 90^\circ$

The sides $AB$ and $BC$ both meet at $B$, so they are the legs. That leaves $AC$ as the hypotenuse, whichever way the triangle is drawn.

Square each leg, add, and take the square root.

$c^2 = 6^2 + 8^2 = 36 + 64 = 100$

So $c = \sqrt{100} = 10$ cm. Adding the legs gives $14$, and $100$ is $c^2$, not $c$.

The theorem adds the squares of the two legs and matches them with the square of the hypotenuse.

$p^2 + q^2 = r^2$

The hypotenuse $r$ goes alone on its own side, and it is squared too.

The square on the hypotenuse has the same area as the two squares on the legs together.

$25 + 144 = 169$

So it covers $169$ square units. Its side, the hypotenuse, would be $\sqrt{169} = 13$ units.

For a leg, subtract the square of the known leg from the square of the hypotenuse.

$b^2 = 13^2 - 5^2 = 169 - 25 = 144$

So $b = \sqrt{144} = 12$. Adding the squares gives $\sqrt{194}$, which is longer than the hypotenuse and so cannot be a leg.

The theorem says $c^2 = a^2 + b^2$. Because $b^2$ is more than zero, $c^2$ is more than $a^2$.

$c^2 > a^2 \text{ and } c^2 > b^2$

A bigger square has a bigger side, so the hypotenuse is longer than each leg, whichever way the triangle is drawn.

Square the legs and add them.

$c^2 = 81 + 144 = 225$

Since $15 \times 15 = 225$, the hypotenuse is $15$ m. Subtracting the squares gives $\sqrt{63}$, which is the method for a leg.

Add the squares of the legs.

$c^2 = 1^2 + 1^2 = 2$

So $c = \sqrt{2}$ units, which is about $1.41$. It is longer than each leg, as a hypotenuse must be, and shorter than the $2$ you get by adding the legs.

Square the legs and add.

$c^2 = 49 + 576 = 625$

Since $25 \times 25 = 625$, the hypotenuse is $25$.

The $10$ is the hypotenuse, so subtract the squares.

$b^2 = 100 - 36 = 64$

So $b = \sqrt{64} = 8$ cm. Subtracting the sides gives $4$, and $64$ is $b^2$, not $b$.

The square on the hypotenuse has the combined area of the other two.

$16 + 20 = 36$

The hypotenuse is the side of that square, so it is $\sqrt{36} = 6$ cm. The $36$ is an area in square centimeters, not a length.

The proof needs a right angle, and the rule can fail without one. A triangle with all three sides $2$ gives

$2^2 + 2^2 = 8, \text{ not } 2^2 = 4$

So apply the rule once you know the triangle has a right angle.