Finding a Missing Side Using the Pythagorean Theorem

What is the Pythagorean Theorem?

The Pythagorean theorem is a rule in mathematics that applies only to a special type of triangle: the right-angled triangle. This is a triangle where one of the three angles is exactly 90 degrees, like the corner of a square. This 90-degree angle is usually marked with a small square symbol.

The three sides of a right-angled triangle have special names. The two sides that form the right angle are called the legs. The third side, which is always the longest and is found opposite the right angle, is called the hypotenuse.

The Pythagorean theorem gives us a formula that connects the lengths of these three sides:

Here, $a$ and $b$ are the lengths of the two shorter legs, and $c$ is the length of the hypotenuse. The small '2' means the number is 'squared' (multiplied by itself). So, the theorem states that if you square the lengths of the two legs and add them together, you get the square of the length of the hypotenuse.

This is incredibly useful because if we know the lengths of any two sides, we can use this formula to find the length of the third.

Finding the Longest Side (the Hypotenuse)

Let's first look at how to find the hypotenuse when we know the two shorter sides. The following resource provides a fantastic, clear guide.

As the article shows, the process is always the same:

1. Label: Identify the two legs ($a$ and $b$) and the hypotenuse ($c$). The side you are looking for is $c$.
2. Substitute: Put the numbers you know into the formula $a^2+b^2=c^2$.
3. Calculate: Square the two known lengths and add them together. This gives you the value of $c^2$.
4. Square Root: Use the square root button (​) on your calculator to find $c$.

Let's walk through an example together.

Example 1: Find the hypotenuse
Imagine a right-angled triangle with legs of length 7 cm and 9 cm.

1. Label: $a=7$, $b=9$, and $c$ is the unknown hypotenuse.
2. Substitute: $7^2+9^2=c^2$
3. Calculate:
   • $7^2=7\times 7=49$
   • $9^2=9\times 9=81$
   • $49+81=130$. So, $c^2=130$.
4. Square Root: $c=\sqrt{130}$. Using a calculator, this is approximately 11.4.

So, the hypotenuse is 11.4 cm long (rounded to one decimal place).

Finding a Shorter Side (a Leg)

What if you are given the hypotenuse and one of the shorter legs? The process is very similar, but with one crucial change. Instead of adding, we subtract.

To find a missing leg (let's say, $b$), we rearrange the formula:

You always subtract the square of the shorter known side from the square of the longest side (the hypotenuse).

The same resource you just read also explains this process perfectly.

The steps for finding a shorter side are:

1. Label: Identify the known leg (e.g., $a$), the unknown leg (e.g., $b$), and the hypotenuse ($c$).
2. Substitute: Put the numbers into the formula $a^2+b^2=c^2$.
3. Calculate: Square the two known lengths.
4. Subtract: Subtract the square of the known leg from the square of the hypotenuse. This gives you the value of the missing leg squared (e.g., $b^2$).
5. Square Root: Use the square root button on your calculator to find the length of the missing leg.

Let's do one final example.

Example 2: Find a shorter side
Imagine a right-angled triangle with a hypotenuse of 26 m and one leg of 10 m.

1. Label: $a=10$, $c=26$, and $b$ is the unknown leg.
2. Substitute: $10^2+b^2=26^2$
3. Calculate:
   • $10^2=10\times 10=100$
   • $26^2=26\times 26=676$
4. Subtract: Now, we rearrange to find $b^2$.
   • $b^2=676-100$
   • $b^2=576$
5. Square Root: $b=\sqrt{576}$. Using a calculator, this is exactly 24.

So, the missing leg is 24 m long.

Conclusion

Great work today. We've tackled the first essential building block for your trigonometry test. You now have a reliable method for solving what used to be a confusing problem.

Here are the key takeaways from today's lesson:

• The Pythagorean theorem, $a^2+b^2=c^2$, is used to find a missing side in a right-angled triangle.
• The side labeled $c$ must always be the hypotenuse (the longest side, opposite the right angle).
• To find the hypotenuse, you ADD the squares of the two shorter sides and then take the square root.
• To find a shorter side, you SUBTRACT the square of the known leg from the square of the hypotenuse, and then take the square root.

In our next lesson, we will build directly on this. We'll learn how to label the sides of a right-angled triangle as "opposite", "adjacent", and "hypotenuse" in relation to a specific angle. This is the next step toward using the main trigonometric functions: sine, cosine, and tangent.