Which equation can be used to find $x$, the length of the hypotenuse of the right triangle?
$16+63=x$
$16^2+63=x$
$(16+63)^2=x^2$
$16^2+63^2=x^2$
Answer (2)
The problem involves finding the correct equation to determine the length of the hypotenuse of a right triangle using the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Given the legs of the right triangle are 16 and 63, we apply the theorem: a 2 + b 2 = x 2 .
Substituting the given values, the equation becomes 1 6 2 + 6 3 2 = x 2 .
Therefore, the correct equation is 1 6 2 + 6 3 2 = x 2 .
Explanation
Problem Analysis and Pythagorean Theorem We are given a right triangle with legs of length 16 and 63, and we want to find the length of the hypotenuse, denoted by x . The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Applying the Pythagorean Theorem The Pythagorean theorem can be written as: a 2 + b 2 = c 2 where a and b are the lengths of the legs, and c is the length of the hypotenuse. In our case, a = 16 , b = 63 , and c = x . Substituting these values into the Pythagorean theorem, we get: 1 6 2 + 6 3 2 = x 2
The Equation Therefore, the equation that can be used to find x is 1 6 2 + 6 3 2 = x 2 .
Examples
The Pythagorean theorem is a fundamental concept in geometry and has many real-world applications. For example, if you are building a ramp that needs to reach a certain height and span a certain horizontal distance, you can use the Pythagorean theorem to calculate the length of the ramp needed. Suppose you want to build a ramp that is 4 feet high and spans a horizontal distance of 10 feet. The length of the ramp (the hypotenuse) can be found using the Pythagorean theorem: 4 2 + 1 0 2 = c 2 , so 16 + 100 = c 2 , which means c 2 = 116 , and c = 116 ≈ 10.77 feet. Therefore, you would need a ramp that is approximately 10.77 feet long.
The equation used to find the length of the hypotenuse of a right triangle with legs of lengths 16 and 63 is 1 6 2 + 6 3 2 = x 2 . This follows from the Pythagorean theorem. Therefore, the correct choice is 1 6 2 + 6 3 2 = x 2 .
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