Consider triangle ABC. The legs have a length of 5 units each.
What is the length of the hypotenuse of the triangle?
A. 5 units
B. [tex]5 \sqrt{2}[/tex] units
C. 10 units
D. [tex]10 \sqrt{2}[/tex] units
Answer (1)
Recognize the problem as finding the hypotenuse of a right triangle.
Apply the Pythagorean theorem: a 2 + b 2 = c 2 .
Substitute the given leg lengths: 5 2 + 5 2 = c 2 .
Solve for the hypotenuse: c = 50 = 5 2 .
The length of the hypotenuse is 5 2 units.
Explanation
Problem Analysis We are given a triangle ABC where the legs have a length of 5 units each. We need to find the length of the hypotenuse of the triangle. Since the problem states that the legs have equal length, we can assume that this is a right triangle.
Apply Pythagorean Theorem Let a and b be the lengths of the legs of the right triangle, and let c be the length of the hypotenuse. We are given that a = 5 and b = 5 . According to the Pythagorean theorem, we have: a 2 + b 2 = c 2
Substitute Values Substitute the given values into the Pythagorean theorem: 5 2 + 5 2 = c 2 25 + 25 = c 2 50 = c 2
Solve for Hypotenuse Solve for c by taking the square root of both sides: c = \[ \sqrt{50} \] c = 25 × 2 c = 5 2
Final Answer The length of the hypotenuse of the triangle is 5 2 units.
Examples
Understanding the length of a hypotenuse is useful in construction and navigation. For example, if you are building a ramp that rises 5 feet over a horizontal distance of 5 feet, the length of the ramp (the hypotenuse) would be 5 2 feet. This ensures you have the correct amount of material and that the ramp meets the required specifications. Another example is in navigation, where you might need to calculate the direct distance between two points that are 5 miles east and 5 miles north of each other; the straight-line distance is the hypotenuse of a right triangle.
