# The Converse of the Pythagorean Theorem

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Chris S.

## The Converse of the Pythagorean Theorem

The converse of the Pythagorean Theorem is an important idea in geometry. It helps us figure out if a triangle has a right angle. If adding the squares of the two shorter sides equals the square of the longest side, then the triangle is a right triangle. Applying the Pythagorean Theorem is very useful for solving math problems and is also used in many real-life situations where right angles are needed.

The Converse of the Pythagorean Theorem states that for any triangle with sides of lengths a, b, and c (where c is the longest side), if a2 + b2 = c2 holds true, then the triangle is a right triangle with the right angle opposite side c.

## The Converse of the Pythagorean Theorem – Explanation

The converse of the Pythagorean Theorem helps confirm the nature of a triangle by relating the squares of its side lengths. This property is especially useful in fields like construction, where confirming right angles is crucial for the integrity of structures.

Step Description Detail
1 Identify the sides Determine which side would be the hypotenuse if the triangle were right-angled. This is always the longest side, $c$.
2 Apply the theorem Square the lengths of the shorter sides ($a$ and $b$) and sum them up.
3 Compare the sum to $c^{2}$ If $a^{2} + b^{2}$ equals $c^{2}$, the triangle is right.

## The Converse of the Pythagorean Theorem – Examples

Let’s practice some together.

Example 1: Triangle with sides 6 units, 8 unit, and 10 units

• Identify the sides: The longest side, which would be the hypotenuse in a right triangle, is $c = 10$ units.
• Apply the theorem: Calculate and sum the squares of the other two sides: $a^{2} = 6^{2} = 36$ and $b^{2} = 8^{2} = 64$. The sum is $a^{2} + b^{2} = 36 + 64 = 100$.
• Compare the sum to $c^{2}$: Since $c^{2} = 10^{2} = 100$ and $a^{2} + b^{2} = c^{2}$, by the converse of the Pythagorean Theorem, the triangle is a right triangle.

Example 2: Triangle with sides 5 units, 12 units, and 13 units

• Identify the sides: The longest side, which would be the hypotenuse in a right triangle, is $c = 13$ units.
• Apply the theorem: Calculate and sum the squares of the other two sides: $a^{2} = 5^{2} = 25$ and $b^{2} = 12^{2} = 144$. The sum is $a^{2} + b^{2} = 25 + 144 = 169$.
• Compare the sum to $c^{2}$: Since $c^{2} = 13^{2} = 169$ and $a^{2} + b^{2} = c^{2}$, by the converse of the Pythagorean Theorem, the triangle is a right triangle.

Let’s apply the converse of the Pythagorean Theorem to determine if a triangle with sides of 5 units, 12 units, and 13 units is a right triangle.

Calculate $5^{2} + 12^{2}$ and compare it to $13^{2}$.

Is the triangle with sides measuring 9 units, 12 units, and the longest side (hypotenuse) measuring 15 units a right triangle? Let’s apply the converse of the Pythagorean Theorem!

Calculate $9^{2} + 12^{2}$ and compare it to $15^{2}$.

Can a triangle with side lengths of 7 units, 24 units, and a hypotenuse of 25 units be classified as a right triangle? Use the Pythagorean Theorem to find out.

Calculate $7^{2} + 24^{2}$ and compare it to $25^{2}$.

## The Converse of the Pythagorean Theorem – Practice

Is the triangle with side lengths of 6 units, 8 units, and 10 units a right triangle?
Is the triangle with side lengths of 3 units, 4 units, and 5 units a right triangle?
Is the triangle with side lengths of 5 units, 12 units, and 13 units a right triangle?
Is the triangle with side lengths of 8 units, 15 units, and 17 units a right triangle?
Is the triangle with side lengths of 9 units, 40 units, and 41 units a right triangle?

## The Converse of the Pythagorean Theorem – Summary

Key Learnings from this Text:

• The converse of the Pythagorean Theorem allows us to verify whether a triangle is a right triangle.
• By verifying that $a^{2} + b^{2} = c^{2}$, where $c$ is the longest side, we can conclusively determine the triangle's right-angle nature.

## The Converse of the Pythagorean Theorem – Frequently Asked Questions

What is the Converse of the Pythagorean Theorem?
How do you determine the hypotenuse in the Converse of the Pythagorean Theorem?
Can the Converse of the Pythagorean Theorem be used for any triangle?
What is a right triangle?
Why is the Converse of the Pythagorean Theorem important in construction?
How can the Converse of the Pythagorean Theorem be visually represented?
Is there a real-world example where the Converse of the Pythagorean Theorem might be used?
What does it mean if $a^{2} + b^{2}$ does not equal $c^{2}$ in a triangle?
Can the Converse of the Pythagorean Theorem be used in navigation?
How does the Converse of the Pythagorean Theorem relate to Pythagorean triples?

## The Converse of the Pythagorean Theorem exercise

Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the video The Converse of the Pythagorean Theorem.
• ### Find the square of the numbers.

Hints

When we square a number we multiply it by itself. A skill we need when we are using the Pythagorean Theorem. For example, $4^2 = 4\times4 = 16$

When we square a decimal with one decimal place the answer will have two decimal places. For example, $0.1^2 = 0.1\times0.1 = 0.01$.

Solution
• $3^2 = 3\times3 = 9$. We can see this above as the area of a square with sides $3$.
• $0.5^2 = 0.5\times0.5 = 0.25$
• $5^2 = 5\times5 = 25$
• $0.2^2 = 0.2\times0.2 = 0.04$
• ### Understanding the converse of the Pythagorean theorem.

Hints

We use the Pythagorean Theorem to find out if the triangle is right angled.

If, $a^2 + b^2$ is the same value as $c^2$ then the triangle is right angled.

Substitute the values into the theorem and square them.

For example, to check this triangle we use:

• $a^2 + b^2 = c^2$
• $2^2 + 3^2 = 4^2$
• $4 + 9 = 16$
• This is not correct as $4 + 9 = 13$ not $16$ therefore, the triangle is not right angled.

Solution
• The Pythagorean Theorem states that $a^2 + b^2 = c^2$
• Using this triangle $7^2 + 40^2 = 42^2$
• We square all the numbers so, $49 + 1600 = 1764$
• Therefore the triangle is not right angled!
• ### Use the converse of the Pythagorean Theorem.

Hints

For a triangle to be right the Pythagorean Theorem should be true. That is $a^2 + b^2 = c^2$, where $c$ is the longest side, the hypotenuse.

For example, if $a = 9$, $b = 12$ and $c = 15$ we can use the Pythagorean theorem to check if the triangle is right.

• $a^2 + b^2 = c^2$
• $9^2 + 12^2 = 15^2$
• $81 + 144 = 225$
• As $81 + 144$ does $= 225$, the triangle is right.
If this was not true then it would not be a right triangle.

There are $2$ correct answers.

Solution

Right triangles are:

• $a = 3, b = 4, c = 5$ as $3^2 + 4^2 = 5^2$ as $9 + 16 = 25$ Correct
• $a = 6, b = 8, c = 10$ as $6^2 + 8^2 = 10^2$ as $36 + 64 = 100$ Correct.
There are two which are not right:
• $a = 4, b = 5, c = 6$ as $4^2 + 5^2 = 6^2$ as $16 + 25 = 36$ Not correct.
• $a = 8, b = 10, c = 12$ as $8^2 + 10^2 = 12^2$ as $64 + 100 = 144$ Not correct.

• ### Use the Pythaogrean theorem to prove which triangle has a right angle.

Hints

For a triangle to be right, the Pythagorean Theorem should be true. That is $a^2 + b^2 = c^2$, where $c$ is the longest side, the hypotenuse.

For example, if $a = 1$, $b = 2$ and $c = 3$ we can use the Pythagorean theorem to check if the triangle is right.

• $a^2 + b^2 = c^2$
• $1^2 + 2^2 = 3^2$
• $1 + 4 = 9$
• As $1 + 4$ does not $= 9$, the triangle is not right.
If this was true then it would be a right triangle.

Solution

The right triangle is:

• $a = 1.5, b = 2, c = 2.5$
If we substitute into the formula

• $1.5^2 + 2^2 = 2.5^2$
• $2.25 + 4 = 6.25$
This is correct and the triangle is right.

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The others are not equal so are not right triangles.

For example,

• $a = 2, b = 4, c = 6$
• $2^2 + 4^2 = 6^2$
• $4 + 16 = 36$. This is not correct, so is not a right triangle.
• ### Identifying the sides of a right triangle.

Hints

Pythagorean Theorem states that the squares off the two shorter sides added together are equal to the square off the hypotenuse.

Pythagorean Theorem is $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.

The hypotenuse is the side opposite the right angle, and is the longest side in a right triangle.

As an example, we can see here that,

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

The Pythagorean Theorem is $a^2 + b^2 = c^2$ where $c$ is the hypotenuse (the longest side, opposite the right angle).

• ### Indirect proof

Hints

When completing an indirect proof of a right triangle we firstly write the statement and assume the triangle is not a right triangle.

We use the triangle to try to prove it is NOT right. That means it is not equal to $a^2 + b^2 = c^2$.

If we find the assumption is not correct, we can say it is a right triangle.

Solution

An indirect proof is when we state the theorem, then try to prove otherwise. If we cannot prove otherwise then we can indirectly say the theorem is correct.