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Pythagorean Theorem and its Converse

Use a-squared plus b-squared equals c-squared to prove triangles are right triangles.

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Pythagorean Theorem and its Converse

Last year, Yuli and his dad made a square bed for vegetables and edged it with timbers. This year his mom wants another one for flowers that measures 5 feet by 5 feet and has a timber diagonally through the middle to separate the sections. How long should the dividing timber be?

In this concept, you will learn how to use the Pythagorean Theorem and its converse.

Using the Pythagorean Theorem and its Converse

You already know that a square consists of four equal sides and four equal, right, angles.

When you divide a square in half by a diagonal, it forms two right triangles.

In the labeled right triangle below:

Sides a\begin{align*}a\end{align*} and b\begin{align*}b\end{align*} are adjacent to the right angle and are called the legs of the triangle.

Side c\begin{align*}c\end{align*}, the longest side, is opposite the right angle and is called the hypotenuse of the triangle.

The Pythagorean Theorem is named for the Greek mathematician Pythagoras, who discovered the unique relationship between the dimensions of a right triangle. The theorem says that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse.

a2+b2=c2\begin{align*}a^2 + b^2 = c^2\end{align*}

Let’s look at an example.

Find the hypotenuse.

First, write the formula, and substitute what you know.

a2+b232+42==c2c2\begin{align*}\begin{array}{rcl} a^2 + b^2 &=& c^2 \\ 3^2 + 4^2 &=& c^2 \end{array}\end{align*}

Next, do the calculations.

9+1625==c2c2\begin{align*}\begin{array}{rcl} 9 + 16 &=& c^2 \\ 25 &=& c^2 \end{array}\end{align*}

Then, take the square root of both sides of the equation.

5=c\begin{align*}5 = c\end{align*}

The answer is 5. The hypotenuse of the triangle is 5.

Here’s another example.

Find the length of the hypotenuse in the triangle below by using the Pythagorean Theorem.

First, write the formula, and substitute.

c2c2==a2+b262+82\begin{align*}\begin{array}{rcl} c^2&=& a^2+b^2 \\ c^2&=& 6^2+8^2 \end{array}\end{align*}

Next, calculate the squares.

c2c2==36+64100\begin{align*}\begin{array}{rcl} c^2&=&36+64 \\ c^2&=& 100 \end{array}\end{align*}

Then take the square root of each side.

The answer is c=10\begin{align*}c=10\end{align*}. The length of the hypotenuse is 10.

Examples

Example 1

Earlier, you were given a problem about Yuli and his garden beds.

Yuli needs to know how long a diagonal timber he needs for a 5 ft×5 ft\begin{align*}5 \ ft \times 5 \ ft\end{align*} square.

First, use the Pythagorean Theorem and substitute in the values you know.

c2c2==a2+b252+52\begin{align*}\begin{array}{rcl} c^2&=& a^2+b^2 \\ c^2&=&5^2+5^2 \end{array}\end{align*}

Next, calculate the value of the squares.

c2c2==25+2550\begin{align*}\begin{array}{rcl} c^2 &=&25+25 \\ c^2&=&50 \end{array}\end{align*}

Then take the square root of each side.

c2=50\begin{align*}\sqrt{c^2}=\sqrt{50}\end{align*}

c=7.071067812\begin{align*}c = 7.071067812 \ldots\end{align*}

Round the number.

c7.07\begin{align*}c\approx 7.07\end{align*}

The answer is that the hypotenuse c7.07\begin{align*}c \approx 7.07\end{align*}, but since Yuli may have a hard time measuring 7100\begin{align*}\frac{7}{100}\end{align*} of a foot, it is safe to say that Yuli can use a 7 ft timber.

Example 2

What is the length of the hypotenuse of this triangle?

First, write the formula and fill in what you know.

72+102=c2\begin{align*}7^2 + 10^2 = c^2\end{align*}

Next, do the calculations.

49+100149==c2c2\begin{align*}\begin{array}{rcl} 49 + 100 &=& c^2 \\ 149 &=& c^2 \end{array}\end{align*}

Then take the square roots.

149=c\begin{align*}\sqrt{149}=c\end{align*}

Since 149 is not a perfect square, use your calculator.

149=12.20655562\begin{align*}\sqrt{149} = 12.20655562 \ldots\end{align*}

Round to the nearest tenth.

The answer is 149=12.2\begin{align*}\sqrt{149}=12.2\end{align*}

Example 3

A right triangle has legs that measure 9 and 12.

What is the hypotenuse?

First, fill the values into the equation.

92+122=c2\begin{align*}9^2+12^2=c^2\end{align*}

Next, perform the calculations.

81+144225==c2c2\begin{align*}\begin{array}{rcl} 81+144&=& c^2 \\ 225&=& c^2 \end{array}\end{align*}

Then, determine the square roots.

15=c\begin{align*}15 = c\end{align*}

The answer is 15. The hypotenuse is 15.

Example 4

Find the hypotenuse of a right triangle with legs of 3 and 6.

a2+b2=c232+62=c2\begin{align*}\begin{array}{rcl} a^2+b^2=c^2 \\ 3^2+6^2=c^2 \end{array}\end{align*}

Next, square the legs.

9+36=c245=c2\begin{align*}\begin{array}{rcl} 9+36=c^2 \\ 45=c^2 \end{array}\end{align*}

Then calculate the square roots of both sides of the equation.

45=c\begin{align*}\sqrt{45}=c\end{align*}

Use a calculator.

c=6.708203933\begin{align*}c = 6.708203933 \ldots\end{align*}

The answer is c6.7\begin{align*}c \approx 6.7\end{align*}. The hypotenuse is approximately 6.7.

Example 5

Claire’s mom needs to fix a broken shingle on their house. If the shingle is 16 feet above the ground and Claire places the foot of the ladder 12 feet from the wall, how long will the ladder need to be?

First, draw a picture to help determine what you are solving for.

In this picture, you can see that the ladder is the hypotenuse.

Next, insert the values of the legs into the formula for the Pythagorean Theorem.

c2c2==a2+b2122+162\begin{align*}\begin{array}{rcl} c^2&=& a^2+b^2 \\ c^2&=& 12^2+16^2 \end{array}\end{align*}

Next, calculate the value of the squares.

c2c2==144+256400\begin{align*}\begin{array}{rcl} c^2&=& 144+256 \\ c^2&=&400 \end{array}\end{align*}

Then take the square root of each side.

The answer is c=20\begin{align*}c=20\end{align*}. The length of the hypotenuse is 10, and the ladder needs to be 10 feet long.

Review

Use the Pythagorean Theorem to find the length of side c\begin{align*}c\end{align*}, the hypotenuse. You may round to the nearest hundredth when necessary.

1. a=6,b=8,c=?\begin{align*}a = 6, b = 8, c =?\end{align*}
2. a=9,b=12,c=?\begin{align*}a = 9, b = 12, c =?\end{align*}
3. a=12,b=16,c=?\begin{align*}a = 12, b = 16, c =?\end{align*}
4. a=15,b=20,c=?\begin{align*}a = 15, b = 20, c =?\end{align*}
5. a=18,b=24,c=?\begin{align*}a = 18, b = 24, c =?\end{align*}
6. a=24,b=32,c=?\begin{align*}a = 24, b = 32, c =?\end{align*}
7. a=5,b=7,c=?\begin{align*}a = 5, b = 7, c =?\end{align*}
8. a=9,b=11,c=?\begin{align*}a = 9, b = 11, c =?\end{align*}
9. a=10,b=12,c=?\begin{align*}a = 10, b = 12, c =?\end{align*}
10. a=12,b=8,c=?\begin{align*}a = 12, b = 8, c =?\end{align*}
11. a=11,b=8,c=?\begin{align*}a = 11, b = 8, c =?\end{align*}
12. a=9,b=8,c=?\begin{align*}a = 9, b = 8, c =?\end{align*}
13. a=15,b=7,c=?\begin{align*}a = 15, b = 7, c =?\end{align*}
14. a=14,b=3,c=?\begin{align*}a = 14, b = 3, c =?\end{align*}
15. a=12,b=8,c=?\begin{align*}a = 12, b = 8, c =?\end{align*}

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Color Highlighted Text Notes

Vocabulary Language: English

TermDefinition
converse If a conditional statement is $p \rightarrow q$ (if $p$, then $q$), then the converse is $q \rightarrow p$ (if $q$, then $p$. Note that the converse of a statement is not true just because the original statement is true.
Hypotenuse The hypotenuse of a right triangle is the longest side of the right triangle. It is across from the right angle.
Legs of a Right Triangle The legs of a right triangle are the two shorter sides of the right triangle. Legs are adjacent to the right angle.
Pythagorean Theorem The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by $a^2 + b^2 = c^2$, where $a$ and $b$ are legs of the triangle and $c$ is the hypotenuse of the triangle.

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