ASVAB Study Guide
Geometry · Lesson 3 of 8

The Pythagorean Theorem

Key rules and formulas, worked examples and 9 practice problems with step-by-step solutions.

1.Pythagorean Theorem

In any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

$$a^2 + b^2 = c^2$$

a and b are the lengths of the two perpendicular legs. c is the length of the hypotenuse (the longest side).

Example

A right triangle has legs measuring 3 cm and 4 cm. Find the length of its hypotenuse.

Show solution

Step 1: Identify the legs: $$a = 3$$, $$b = 4$$. Set up the formula: $$c^2 = 3^2 + 4^2$$

Step 2: Calculate the squares and add them together: $$c^2 = 9 + 16 = 25$$

Step 3: Take the square root of the sum to find c: $$c = \sqrt{25} = 5$$ cm

Practice problems

  1. Problem 1

    A construction worker places a ladder against a wall. The base of the ladder is 6 feet from the wall, and it reaches 8 feet up the wall. How long is the ladder?

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    Answer: 10 feet

    Step 1: Identify the two perpendicular legs: $$a = 6$$ ft, $$b = 8$$ ft

    Step 2: Set up the equation for the hypotenuse: $$c^2 = 6^2 + 8^2 = 36 + 64 = 100$$

    Step 3: Take the square root to find the length: $$c = \sqrt{100} = 10$$ feet

  2. Problem 2

    A right triangular plot of land has legs that measure 5 meters and 12 meters. Calculate the length of the diagonal boundary line (hypotenuse).

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    Answer: 13 meters

    Step 1: Identify the given side parameters: $$a = 5$$, $$b = 12$$

    Step 2: Square both sides and sum them up: $$c^2 = 5^2 + 12^2 = 25 + 144 = 169$$

    Step 3: Solve for c by calculating the square root: $$c = \sqrt{169} = 13$$ meters

  3. Problem 3

    An screen monitor measures 9 inches horizontally and 12 inches vertically. What is the diagonal measurement of the screen?

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    Answer: 15 inches

    Step 1: The horizontal and vertical lines act as the legs: $$a = 9$$, $$b = 12$$

    Step 2: Apply the theorem formula: $$c^2 = 9^2 + 12^2 = 81 + 144 = 225$$

    Step 3: Find the square root of the total value: $$c = \sqrt{225} = 15$$ inches

Remember

Always ensure you isolate the hypotenuse as the side opposite the 90-degree right angle before applying the formula.

2.Finding a Leg

Calculating the length of an unknown leg in a right triangle when the hypotenuse and one leg are known.

$$a^2 = c^2 - b^2$$ (or $$b^2 = c^2 - a^2$$)

c is the known hypotenuse. a and b represent the shorter perpendicular sides of the right triangle.

Example

A right triangle has a hypotenuse of 10 cm and one leg of 6 cm. Find the length of the missing leg.

Show solution

Step 1: Identify the known values: $$c = 10$$, $$b = 6$$. Set up the equation: $$a^2 = 10^2 - 6^2$$

Step 2: Compute the squares and perform subtraction: $$a^2 = 100 - 36 = 64$$

Step 3: Take the square root of the remaining value: $$a = \sqrt{64} = 8$$ cm

Practice problems

  1. Problem 4

    A 15-foot ramp spans from the ground to a loading dock platform. If the horizontal distance along the ground is 12 feet, how high is the loading dock?

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    Answer: 9 feet

    Step 1: Identify the hypotenuse $$(c = 15)$$ and the known horizontal leg $$(b = 12)$$

    Step 2: Subtract the squared leg from the squared hypotenuse: $$a^2 = 15^2 - 12^2 = 225 - 144 = 81$$

    Step 3: Take the square root to calculate the vertical height leg: $$a = \sqrt{81} = 9$$ feet

  2. Problem 5

    The hypotenuse of a right triangle is 25 inches long, and one of its legs measures 7 inches. What is the length of the other leg?

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    Answer: 24 inches

    Step 1: Note down the parameters given: $$c = 25$$, $$a = 7$$

    Step 2: Isolate the missing leg variable value: $$b^2 = 25^2 - 7^2 = 625 - 49 = 576$$

    Step 3: Solve by calculating the square root: $$b = \sqrt{576} = 24$$ inches

  3. Problem 6

    A right triangle has a longest side measuring 26 cm. If one of the shorter perpendicular sides is 10 cm, find the other side.

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    Answer: 24 cm

    Step 1: Set up the formula for finding the leg: $$\text{side}^2 = 26^2 - 10^2$$

    Step 2: Solve the exponential calculations: $$\text{side}^2 = 676 - 100 = 576$$

    Step 3: Isolate the target variable: $$\text{side} = \sqrt{576} = 24$$ cm

Remember

Always subtract the square of the known leg from the square of the hypotenuse; never add them.

3.Converse Theorem

Determining if a given triangle is a right triangle by checking if the square of the longest side equals the sum of the squares of the two shorter sides.

If $$a^2 + b^2 = c^2$$, then the triangle is a right triangle.

a and b are the two shorter side inputs. c must always represent the longest side value checked.

Example

Determine if a triangle with side lengths of 7 cm, 24 cm, and 25 cm is a right triangle.

Show solution

Step 1: Isolate the longest side as c (25) and the shorter sides as a (7) and b (24)

Step 2: Compare the squares: $$7^2 + 24^2 = 49 + 576 = 625$$. Then check $$c^2$$: $$25^2 = 625$$

Step 3: Since 625 equals 625, conclude that it is a right triangle

Practice problems

  1. Problem 7

    A technician measures a triangular framework with side dimensions of 8 inches, 15 inches, and 17 inches. Is this a right triangle?

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    Answer: Yes, it is a right triangle

    Step 1: Set the maximum value as the potential hypotenuse: $$c = 17$$. The shorter sides are $$a = 8$$, $$b = 15$$

    Step 2: Add the squares of the legs: $$8^2 + 15^2 = 64 + 225 = 289$$. Calculate $$c^2$$: $$17^2 = 289$$

    Step 3: Because $$289 = 289$$, the sides form a right triangle

  2. Problem 8

    A triangle has measured side boundaries of 4 cm, 5 cm, and 7 cm. Does this create a right triangle?

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    Answer: No, it is not a right triangle

    Step 1: Assign values: $$a = 4$$, $$b = 5$$, and the longest side $$c = 7$$

    Step 2: Calculate the values: $$4^2 + 5^2 = 16 + 25 = 41$$. Check $$c^2$$: $$7^2 = 49$$

    Step 3: Since 41 does not equal 49, it cannot be a right triangle

  3. Problem 9

    Test whether a triangle with side records of 9 meters, 40 meters, and 41 meters satisfies the requirements of a right triangle.

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    Answer: Yes, it is a right triangle

    Step 1: Identify parameters: $$a = 9$$, $$b = 40$$, and $$c = 41$$

    Step 2: Evaluate both sides of the theorem: $$9^2 + 40^2 = 81 + 1600 = 1681$$. Calculate $$c^2$$: $$41^2 = 1681$$

    Step 3: Confirm that $$1681 = 1681$$, verifying it is a valid right triangle

Remember

If the calculation results in an unequal equation ($$a^2 + b^2 \neq c^2$$), the tested triangle is not a right triangle.