8.1 Pythagorean Theorem and its converse

The Pythagorean Theorem

The Pythagorean theorem gives you a way to find any missing side length in a right triangle when you know the other two. It also works in reverse: you can use its converse to test whether a triangle is a right triangle just from its side lengths.

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Pythagorean Theorem for Side Lengths

In any right triangle, the square of the hypotenuse equals the sum of the squares of the two legs:

$a^2 + b^2 = c^2$

• $c$ is always the hypotenuse (the longest side, opposite the right angle)
• $a$ and $b$ are the two legs (the sides that form the right angle)

Finding a missing hypotenuse: Add the squares of the legs, then take the square root.

Finding a missing leg: Subtract the square of the known leg from the square of the hypotenuse, then take the square root.

A common mistake is forgetting that $c$ must be the longest side. If you accidentally assign a leg as $c$, you'll get a negative value under the radical, which signals something went wrong.

Converse of the Pythagorean Theorem

The converse flips the logic: if the side lengths of a triangle satisfy $a^2 + b^2 = c^2$, then the triangle is a right triangle.

To test whether three given side lengths form a right triangle:

1. Identify the longest side and call it $c$.
2. Square all three sides.
3. Check whether $a^2 + b^2 = c^2$.
   • If equal, the triangle is a right triangle.
   • If not equal, it is not a right triangle.

For Honors, note the two inequality cases as well:

• If $a^2 + b^2 > c^2$, the triangle is acute (all angles less than 90°).
• If $a^2 + b^2 < c^2$, the triangle is obtuse (the angle opposite $c$ is greater than 90°).

This means you can classify any triangle by type using only its side lengths.

Real-World Applications

Many real-world problems contain a hidden right triangle. The key is recognizing it.

Problem-solving steps:

1. Sketch the situation and identify the right triangle.
2. Label the known sides and determine which side is unknown.
3. Apply $a^2 + b^2 = c^2$ and solve.

Ladder against a wall: A 13-ft ladder leans against a wall with its base 5 ft from the wall. How high up the wall does it reach? The ladder is the hypotenuse, so $h = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$ ft.

Distance formula on the coordinate plane: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ comes directly from the Pythagorean theorem. The horizontal and vertical distances are the legs, and the straight-line distance is the hypotenuse:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

3D diagonal of a rectangular prism: To find the space diagonal of a box with length $l$, width $w$, and height $h$, you extend the theorem into three dimensions:

$d = \sqrt{l^2 + w^2 + h^2}$

This works because you're applying the Pythagorean theorem twice: once across the base, then again from that diagonal up through the height.