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Right Triangle Hypotenuse Length Calculation with Pythagorean Theorem
Mathematics
Junior High School (Grade 7-8)
Question Content
In a right triangle, the lengths of the two legs are 5 cm and 12 cm. What is the length of the hypotenuse? \nA) 13 cm \nB) 14 cm \nC) 15 cm \nD) 17 cm
Correct Answer
13 cm
Detailed Solution Steps
1
Step 1: Recall the Pythagorean theorem for a right triangle: \( c = \sqrt{a^2 + b^2} \), where \( a \) and \( b \) are the lengths of the legs, and \( c \) is the hypotenuse.
2
Step 2: Substitute \( a = 5 \) cm and \( b = 12 \) cm into the formula: \( c = \sqrt{5^2 + 12^2} \).
3
Step 3: Calculate the squares: \( 5^2 = 25 \) and \( 12^2 = 144 \).
4
Step 4: Add the results: \( 25 + 144 = 169 \).
5
Step 5: Take the square root: \( \sqrt{169} = 13 \). So the hypotenuse length is 13 cm.
Knowledge Points Involved
1
Pythagorean Theorem
In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (legs). The formula is \( c^2 = a^2 + b^2 \), where \( c \) is the hypotenuse, and \( a, b \) are the legs. It is fundamental in geometry for finding side lengths in right triangles.
2
Square and Square Root Calculation
Squaring a number (e.g., \( 5^2 = 25 \)) means multiplying it by itself. The square root of a perfect square (e.g., \( \sqrt{169} = 13 \)) is the number that, when squared, gives the original number. These operations are used to solve for unknown sides in geometric theorems like the Pythagorean theorem.
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