勾股定理的逆定理

Converse of the Pythagorean Theorem

Don't let the name of this concept stress you out. It's not as complicated as it might seem.

Recall that the Pythagorean Theorem states, for a right triangle with legs of length $a$ and $b$ and hypotenuse of length $c$, that $a^2+b^2=c^2$. The hypotenuse is the side that is across from the right angle, and it is the longest side of the triangle. $a$ and $b$ are the lengths of the other two sides, called legs.

The Converse of the Pythagorean Theorem tells us that if the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

So, what the Converse of the Pythagorean Theorem allows us to do is to determine if a triangle is a right triangle.

If you plug in the lengths of each side of the triangle into the Pythagorean equation and get a TRUE statement, then the triangle is a right triangle.

Note: In order for $a^2+b^2=c^2$ to be true, the side length $c$ must be the largest side length. So in order to test whether a triangle is right, you only need to plug in values where $c$ is the largest among $a,b,c$.

$$3^2+4^2=5^2$$ $$9+16=25$$ $$25=25$$

This produces a true statement ($25=25$), which tells us that our triangle is in fact a right triangle.

The order of $3$ and $4$ (as $a$ and $b$) doesn't really matter. However, it is important that we checked the Pythagorean rule by plugging in $c=5$, the largest among $3,4,5$. If we had plugged in $c=4$ or $c=3$, we would've gotten something like $5^2+3^2\neq 4^2$ or $4^2+5^2\neq 3^2$ instead. The test is only useful when $c$ is the largest side.

$$4^2+4^2=6^2$$ $$16+16=36$$ $$32\neq 36$$

We can tell that this triangle is not a right triangle because the result of using the Pythagorean equation produces a FALSE statement, even though we plugged in the largest side length, $6$, for $c$.

The Converse of the Pythagorean Theorem

Lets begin by asking the question: "What is a converse?"

In mathematics a converse is a theorem whose hypothesis and conclusion are the conclusion and hypothesis of another theorem. In other words, the converse of the Pythagorean Theorem is the reversal of the Pythagorean Theorem.

The Pythagorean Theorem states that the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse: $$a^2+b^2=c^2$$ The converse of the Pythagorean Theorem is the Pythagorean Theorem flipped.

The converse of the Pythagorean Theorem states that: $$\text{If the square of the longest side of the triangle is equal to}$$ $$\text{the sum of the squares of the other two sides of the triangle,}$$ $$\text{then the triangle is a right triangle.}$$

The Pythagorean theorem is used to find the length of a missing side of a right triangle, the converse of the Pythagorean Theorem is used to determine if a triangle is a right triangle or not.

Try it!

Proof of the Converse of the Pythagorean Theorem

The converse of Pythagorean Theorem states that if three sides of a triangle are $a$, $b$ and $c$ such that $a^2 + b^2 = c^2$, then the triangle is right angled.

Proof:

There are many proofs for this but you can use Pythagorean theorem to prove this.

Let the triangle $ABC$ have side lengths $a$, $b$ and $c$ such that $a^2+b^2=c^2$. We will prove that this triangle is right angled.

We will construct a right angled triangle $DEF$ with side lengths $a$, $b$ and $d$ (with $d$ being the hypotenuse). Thus, $a^2 + b^2 = d^2$.

But we already know that $a^2 + b^2 = c^2$. This implies $c^2 = d^2$, which implies $c=d$. Therefore, triangle $ABC$ is congruent to triangle $DEF$ by the SSS criterion.

Therefore, triangle $DEF$ is right angled.

(Video) Video’s Title

by Youtube Name

In this video, we learn that the converse pythagorean theorem can be used to tell if a triangle is a right triangle given all the side lengths.

$\large{\text{Summary}}$

Pythagorean theorem: If a triangle is a right triangle (has a right angle), then $a^2+b^2=c^2$.

Converse: If $a^2+b^2=c^2$ in a triangle with $c$ is the longest side, then a triangle is a right triangle.

If a triangle is not a right triangle, there are $2$ other options for types of triangles.

1. Acute: Has $3$ acute (less than $90$ degree) angles.

2. Obtuse: Has a single obtuse (greater than $90$ degree) angle and $2$ acute angles.

3. Right: Has $1$ right angle ($90$ degrees).

Actue triangles follow the inequality of $c^2<a^2+b^2$. Obtuse triangles follow the inequality of $c^2>a^2+b^2$. Notice, when given a problem, triangles are not always drawn to scale. That means do the math, don't rely on the picture to figure out if a triangle is right!