The length of the hypotenuse in a right triangle is equal to the:
A. Square of the sum of the two other sides (legs)
B. Square root of the sum of the squares of the two other sides (legs)
C. Square root of the square of the sum of the two other sides (legs)
D. Sum of the squares of the two other sides (legs)
Answer (2)
The problem requires recalling the Pythagorean theorem.
The Pythagorean theorem states: c 2 = a 2 + b 2 , where c is the hypotenuse and a and b are the legs of a right triangle.
Solving for c , we get c = a 2 + b 2 .
Therefore, the length of the hypotenuse is the square root of the sum of the squares of the two other sides (legs). Square root of the sum of the squares of the two other sides (legs)
Explanation
Analyze the question Let's analyze the question. We are asked to identify the correct formula for the length of the hypotenuse in a right triangle, given four options. We need to recall the Pythagorean theorem to determine the correct answer.
Apply the Pythagorean theorem The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse ( c ) is equal to the sum of the squares of the lengths of the other two sides (legs, a and b ). This can be written as: c 2 = a 2 + b 2 To find the length of the hypotenuse c , we take the square root of both sides of the equation: c = a 2 + b 2 This means the length of the hypotenuse is the square root of the sum of the squares of the two other sides (legs).
Compare with the given options Now, let's compare this result with the given options:
Square of the sum of the two other sides (legs): ( a + b ) 2 = a 2 + 2 ab + b 2 - Incorrect
Square root of the sum of the squares of the two other sides (legs): a 2 + b 2 - Correct
Square root of the square of the sum of the two other sides (legs): ( a + b ) 2 = a + b - Incorrect
Sum of the squares of the two other sides (legs): a 2 + b 2 - Incorrect
Identify the correct answer The correct option is the 'Square root of the sum of the squares of the two other sides (legs)'.
Examples
The Pythagorean theorem is a fundamental concept in geometry and has numerous real-world applications. For example, architects and engineers use it to ensure that buildings are square and stable. If you're building a ramp, you can use the Pythagorean theorem to calculate the length of the ramp needed to reach a certain height, given the horizontal distance. This ensures the ramp is safe and meets the required specifications.
The length of the hypotenuse in a right triangle is given by the formula c = a 2 + b 2 , according to the Pythagorean theorem. The correct choice from the options provided is B: the square root of the sum of the squares of the two other sides (legs).
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