The legs of a 45-45-90 triangle have a length of 8 units. What is the length of its hypotenuse?
A. [tex]$4 \sqrt{2}$[/tex] units
B. 4 units
C. 8 units
D. [tex]$8 \sqrt{2}$[/tex] units
Answer (1)
Recognize the 45-45-90 triangle and its properties.
Apply the relationship: h y p o t e n u se = l e g × 2 .
Substitute the given leg length: h y p o t e n u se = 8 × 2 .
Determine the hypotenuse length: 8 2 units.
Explanation
Problem Analysis We are given a 45-45-90 triangle with both legs having a length of 8 units. Our goal is to find the length of the hypotenuse.
Key Relationship In a 45-45-90 triangle, there's a special relationship between the lengths of the legs and the hypotenuse. If the legs have length a , then the hypotenuse has length a \[ s q r t 2 . In our case, a = 8 units.
Calculation Now, we can substitute the given leg length into the formula:
h y p o t e n u se = 8 × 2
Calculating this value, we get:
h y p o t e n u se = 8 2 ≈ 11.31 units
Final Answer Therefore, the length of the hypotenuse is 8 2 units. Looking at the multiple-choice options, we see that option D matches our result.
Examples
Imagine you're building a square-shaped garden and want to divide it diagonally to create two equal right-triangular sections. If each side of the square (which becomes the leg of the triangle) is 8 feet long, then the diagonal (the hypotenuse) will be 8 2 feet long. This helps you determine the length of fencing or pathway needed for the diagonal division.
