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Similar Figure and Pythagorean Theorem Word Problems Study Guide (page 2)

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Pythagorean Theorem

The Pythagorean theorem is a special relationship between the sides of any right triangle. This theorem states that the sum of the square of the legs of the triangle is equal to the square of the hypotenuse. Take the following triangle labeled. The legs are labeled a and b and the hypotenuse is labeled c.

Tip:

When you are working with the sides of a right triangle, use the Pythagorean theorem, or a2+ b2= c2.

Look at the following examples. In the first example, the two legs are given, and the hypotenuse is unknown. In the second example, one leg and the hypotenuse are given, and the other leg is unknown.

Example 1

The legs of a right triangle measure 3 m and 4 m. What is the measure of the hypotenuse?

Read and understand the question. This question is looking for the hypotenuse of a right triangle when the lengths of the two legs are known.

Make a plan. Use the formula a2+ b2= c2and substitute the given values. Then, solve the equation for the unknown value.

Carry out the plan. The legs are 3 m and 4 m, so a = 3 and b = 4. Use the formula a2+ b2= c2and substitute the given values: 32+ 42= c2. Evaluate the exponents: 9 + 16 = c2. Add: 25 = c2. Take the positive square root of each side of the equation: 5 = c. The length of the hypotenuse is 5 m.

Check your answer. To check this answer, substitute the lengths of the three sides into the formula. The formula a2+ b2= c2becomes 32+ 42= 52. Apply the exponents to get 9 + 16 = 25. Add the numbers on the left side to get 25 = 25. This answer is checking.

Example 2

The leg of a right triangle measures 24 in. and the hypotenuse measures 26 in. What is the measure of the other leg of the triangle?

Read and understand the question. This question is looking for a leg of a right triangle when the lengths of the other leg and the hypotenuse are known.

Make a plan. Use the formula a2+ b2= c2and substitute the given values. Then, solve the equation for the unknown value.

Carry out the plan. One leg is 24 in. and the hypotenuse is 26 in., so a = 24 and c = 26. Use the formula a2+ b2= c2and substitute the given values: 242+ b2= 262. Evaluate the exponents: 576 + b2= 676. Subtract 576 from each side of the equation to get the variable alone.

576 – 576 + b2 = 676 – 576
b2= 100

Take the positive square root of each side of the equation: b = 10. The length of the other leg is 10 in.

Check your answer. To check this answer, substitute the lengths of the three sides into the formula. The formula a2+ b2= c2becomes 102+ 242= 262. Apply the exponents to get 100 + 576 = 676. Add the numbers on the left side to get 676 = 676. This answer is checking.

Find practice problems and solutions for these concepts at Similar Figure and Pythagorean Theorem Word Problems Practice Questions.

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