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GED Math · Geometry

The Third Side

The Pythagorean theorem — a worksheet

What this is about

When you know two sides of a right triangle, you can find the third

A right triangle is a triangle with one square corner — a 90° angle. The little square drawn in the corner is how a test tells you the angle is exactly 90°.

The two sides that meet at the square corner are called the legs. The side across from the square corner is called the hypotenuse. The hypotenuse is always the longest side of a right triangle.

Find the hypotenuse first, every time. Before you do any arithmetic, look for the square corner and then look straight across from it. That side is the hypotenuse, and in the formula it is the letter c. The other two sides are a and b, and it does not matter which one you call which. Most mistakes on this topic are not arithmetic mistakes. They are mistakes about which side is c.

The rule is this:

a² + b² = c²

In words: square one leg, square the other leg, add them, and you have the square of the hypotenuse. Then take the square root to get the hypotenuse itself.

This formula is printed on the GED formula sheet, so you do not have to memorize it. What you do have to know is when to reach for it and which side is c.

Squares you will keep meeting

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Worked example 1

Both legs known — find the hypotenuse

A right triangle has legs of 6 and 8. How long is the hypotenuse?

Step 1. Name the sides. The two given sides meet at the square corner, so they are the legs: a = 6 and b = 8. The hypotenuse c is what we want.

Step 2. Put the numbers into the formula.

6² + 8² = c²

Step 3. Square each leg and add.

36 + 64 = c²
100 = c²

Step 4. Undo the square. If c squared is 100, then c is the square root of 100.

Answer c = √100 = 10

Notice that the answer, 10, is longer than either leg. It should be. If your hypotenuse comes out shorter than one of the legs, something went wrong.

Worked example 2

Hypotenuse known — find the missing leg

A right triangle has a hypotenuse of 13 and one leg of 5. How long is the other leg?

Step 1. Name the sides. This time the 13 is across from the square corner, so 13 is c. The 5 is a leg. The missing side is the other leg.

Step 2. Put in what you know.

5² + b² = 13²
25 + b² = 169

Step 3. Subtract, do not add. Here is the difference between this kind of problem and the first kind. The hypotenuse is already on one side of the equation, so to get b² by itself you take the 25 away from both sides.

b² = 169 − 25 = 144

Step 4. Undo the square.

Answer b = √144 = 12

Add or subtract? Ask one question: is the side I am looking for the hypotenuse, or a leg? Looking for the hypotenuse — add the two squares. Looking for a leg — subtract the leg's square from the hypotenuse's square. That single question is most of this topic.

When the root is not whole. Most real measurements do not come out even. If you get c² = 45, then c = √45, which is about 6.7. Use the calculator, and round the way the question tells you to. If it does not say, one decimal place is a safe choice.

Part A

Find the hypotenuse

Both legs are given. Round to one decimal place where the answer is not a whole number.

  1. Legs 3 and 4.
  2. Legs 9 and 12.
  3. Legs 5 and 12.
  4. Legs 7 and 10.
Part B

Find the missing leg

The hypotenuse and one leg are given. Round to one decimal place where the answer is not a whole number.

  1. Hypotenuse 17, one leg 8.
  2. Hypotenuse 25, one leg 7.
  3. Hypotenuse 10, one leg 6.
  4. Hypotenuse 20, one leg 11.
Part C

Where it shows up

Each of these is a right triangle in disguise. Before you calculate, decide which side is the hypotenuse.

  1. A ladder leans against a building. Its foot is 5 feet out from the wall, and the ladder is 13 feet long. How far up the wall does the ladder reach?
  2. You walk 3 blocks east and then 4 blocks north. If you could walk straight from where you started to where you are now, how many blocks would that be?
  3. A television screen measures 24 inches across and 18 inches high. Screens are sold by the diagonal. What size screen is it?
  4. A rectangular lot is 40 meters by 30 meters. A footpath runs corner to corner across it. How long is the path?
  5. A wheelchair ramp rises 7 feet over a horizontal run of 24 feet. How long is the sloping surface of the ramp?
Part D

Is it a right triangle?

The theorem works in both directions. If the two shorter sides squared and added equal the longest side squared, the triangle has a square corner. If they do not, it does not. Test each set of three sides and write yes or no.

  1. 9, 40, 41
  2. 5, 6, 8
  3. 8, 15, 17
  4. 10, 12, 16
Answer key — open when you have finished

Part A — find the hypotenuse

1. 5. 9 + 16 = 25, and √25 = 5. This 3-4-5 triangle turns up constantly; it is worth knowing by sight.
2. 15. 81 + 144 = 225, and √225 = 15. It is the 3-4-5 triangle with every side tripled.
3. 13. 25 + 144 = 169, and √169 = 13.
4. About 12.2. 49 + 100 = 149. There is no whole number whose square is 149 — it sits between 12² = 144 and 13² = 169 — so √149 ≈ 12.2.

Part B — find the missing leg

1. 15. 289 − 64 = 225, and √225 = 15.
2. 24. 625 − 49 = 576, and √576 = 24.
3. 8. 100 − 36 = 64, and √64 = 8. Another 6-8-10 triangle, the same shape as the worked example.
4. About 16.7. 400 − 121 = 279, and √279 ≈ 16.7. Check it against the hypotenuse: 16.7 is less than 20, as a leg must be.

Part C — where it shows up

1. 12 feet. The ladder is the hypotenuse, because the square corner is where the wall meets the ground and the ladder lies across from it. So 5² + b² = 13² → b² = 169 − 25 = 144 → b = 12.
2. 5 blocks. East and north meet at a square corner, so those two walks are the legs: 3² + 4² = 9 + 16 = 25, and √25 = 5.
3. 30 inches. The diagonal is the hypotenuse of a right triangle whose legs are the width and the height: 24² + 18² = 576 + 324 = 900, and √900 = 30.
4. 50 meters. 40² + 30² = 1600 + 900 = 2500, and √2500 = 50.
5. 25 feet. The rise and the run are the legs; the sloping surface is the hypotenuse. 7² + 24² = 49 + 576 = 625, and √625 = 25.

Part D — is it a right triangle?

1. Yes. 81 + 1600 = 1681, and 41² = 1681. They match.
2. No. 25 + 36 = 61, but 8² = 64. Close, but not equal — and close does not count.
3. Yes. 64 + 225 = 289, and 17² = 289.
4. No. 100 + 144 = 244, but 16² = 256.