Pythagorean Theorem Proofs and Converse

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse ($c$) is equal to the sum of the squares of the other two sides ($a$ and $b$). Mathematically, this is written as $a^2 + b^2 = c^2$. In this lesson, we will look at how to formally prove this theorem and how to use its converse to classify any triangle.

Proving the Pythagorean Theorem Using Similar Triangles

There are hundreds of ways to prove the Pythagorean theorem, but one of the most elegant methods uses similar triangles.

Imagine a right triangle $\triangle ABC$ with the right angle at $C$. Draw an altitude from $C$ to the hypotenuse $AB$, meeting $AB$ at point $D$. This altitude divides the hypotenuse $c$ into two segments, let's call them $x$ and $y$ (so $c = x + y$).

This creates three similar triangles: the large triangle $\triangle ABC$, and the two smaller triangles $\triangle ACD$ and $\triangle CBD$.

Because the triangles are similar, the ratio of their corresponding sides is equal:

1. From $\triangle ACD \sim \triangle ABC$, we get: $\frac{b}{x} = \frac{c}{b} \implies b^2 = cx$

2. From $\triangle CBD \sim \triangle ABC$, we get: $\frac{a}{y} = \frac{c}{a} \implies a^2 = cy$

Now, add the two equations together: $a^2 + b^2 = cy + cx$ $a^2 + b^2 = c(y + x)$

Since $x + y = c$, we substitute $c$ back into the equation: $a^2 + b^2 = c(c)$ $a^2 + b^2 = c^2$

This completes the proof!

The Converse of the Pythagorean Theorem

The converse of the Pythagorean theorem allows us to work backward. It states: If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

If $c$ is the longest side of a triangle and $a^2 + b^2 = c^2$, then the angle opposite side $c$ is a right angle ($90^\circ$).

Classifying Triangles: Acute, Right, or Obtuse

We can extend the converse to determine whether any given triangle is acute, right, or obtuse just by looking at its side lengths. Let $c$ always represent the longest side of the triangle, and $a$ and $b$ represent the two shorter sides.

First, ensure the sides can actually form a triangle using the Triangle Inequality Theorem ($a + b > c$). If they do, compare $c^2$ to $a^2 + b^2$:

• Right Triangle: If $c^2 = a^2 + b^2$
• Acute Triangle: If $c^2 < a^2 + b^2$ (The longest side is relatively short, closing the angle to less than $90^\circ$)
• Obtuse Triangle: If $c^2 > a^2 + b^2$ (The longest side is stretched out, opening the angle to more than $90^\circ$)

Example Problem

Question: Determine if a triangle with side lengths $5, 11,$ and $13$ is acute, right, or obtuse.

Step 1: Identify the longest side, $c$. Here, $c = 13$. The other two sides are $a = 5$ and $b = 11$.

Step 2: Calculate $c^2$. $c^2 = 13^2 = 169$

Step 3: Calculate $a^2 + b^2$. $a^2 + b^2 = 5^2 + 11^2$ $a^2 + b^2 = 25 + 121 = 146$

Step 4: Compare the values. We see that $169 > 146$, which means $c^2 > a^2 + b^2$.

Conclusion: Because the square of the longest side is greater than the sum of the squares of the other two sides, the triangle is obtuse.