Each leg of a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle measures 12 cm. What is the length of the hypotenuse?
A. 6 cm
B. $5 \sqrt{2} cm$
C. 12 cm
D. $12 \sqrt{2} cm$
Answer (2)
The length of the hypotenuse in a 4 5 ° − 4 5 ° − 9 0 ° triangle with legs measuring 12 cm is 12 √ ( 2 ) cm. Therefore, the correct answer is D: 12 √ ( 2 ) cm.
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Recognize that in a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle, the legs are equal, and apply the Pythagorean theorem.
Substitute the given leg length (12 cm) into the Pythagorean theorem: 1 2 2 + 1 2 2 = c 2 .
Simplify the equation to find c 2 = 288 .
Solve for c to find the hypotenuse length: 12 2 cm .
Explanation
Problem Analysis We are given a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle where each leg measures 12 cm. Our goal is to find the length of the hypotenuse.
Applying the Pythagorean Theorem In a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle, the two legs are of equal length. Let's denote the length of each leg as a , which is given as 12 cm. Let's denote the length of the hypotenuse as c . According to the Pythagorean theorem, the relationship between the legs and the hypotenuse is: a 2 + a 2 = c 2
Substitution and Simplification Substitute the given value of a = 12 cm into the equation: 1 2 2 + 1 2 2 = c 2 144 + 144 = c 2 288 = c 2
Solving for the Hypotenuse Now, solve for c by taking the square root of both sides: c = 288 c = 144 × 2 c = 12 2 cm
Final Answer Therefore, the length of the hypotenuse is $12
\sqrt{2}$ cm.
Examples
Right triangles are fundamental in construction and navigation. For example, when building a ramp that needs to meet a specific angle, understanding the relationship between the sides of a right triangle ensures the ramp is safe and functional. In navigation, right triangles are used to calculate distances and directions, especially when dealing with angles and bearings.
