Problem 37
Question
Determine whether the given lengths are sides of a right triangle. Explain your reasoning. $$ 5,12,13 $$
Step-by-Step Solution
Verified Answer
Yes, the lengths 5, 12, and 13 can form a right triangle.
1Step 1: Identify the longest side
First, identify the longest side among the three given lengths, which will potentially be the hypotenuse of the right triangle. In this case, 13 is the longest side.
2Step 2: Apply Pythagorean Theorem
Next, apply the Pythagorean theorem, which states that the square of the hypotenuse should be equal to the sum of the squares of the other two sides. Therefore, compute \(5^2 + 12^2\) and compare it to \(13^2\).
3Step 3: Compare the Results
Compare the results from Step 2. If they are equal, then the triangle is a right triangle. Otherwise, it is not. In this case, as \(5^2 + 12^2 = 169\) and \(13^2 = 169\) are equal, hence the lengths 5, 12 and 13 could form a right triangle.
Key Concepts
Right Triangle IdentificationApplying Pythagorean TheoremMathematical Reasoning
Right Triangle Identification
Understanding how to identify a right triangle is fundamental in geometry. A right triangle is a type of triangle that has one angle measuring exactly 90 degrees.
Identifying a right triangle involves inspecting its sides and applying certain principles, most notably the Pythagorean theorem. This theorem only holds true for right triangles, and it states that the square of the longest side (the hypotenuse) is equal to the sum of the squares of the other two shorter sides (the legs). So to confirm if a given set of three lengths can form a right triangle, check for this relationship. Remember, the hypotenuse is always opposite the right angle and the longest side of the triangle.
Identifying a right triangle involves inspecting its sides and applying certain principles, most notably the Pythagorean theorem. This theorem only holds true for right triangles, and it states that the square of the longest side (the hypotenuse) is equal to the sum of the squares of the other two shorter sides (the legs). So to confirm if a given set of three lengths can form a right triangle, check for this relationship. Remember, the hypotenuse is always opposite the right angle and the longest side of the triangle.
Applying Pythagorean Theorem
To apply the Pythagorean theorem effectively, first ensure that the sides are labeled correctly. The two shorter sides are called 'legs' and the longest side is the 'hypotenuse'. The formula for the Pythagorean theorem is \( a^2 + b^2 = c^2 \), where \( c \) represents the length of the hypotenuse and \( a \) and \( b \) are the lengths of the other two sides.
For our specific problem, where the sides are 5, 12, and 13, we see that 13 is the longest side and is thus potentially the hypotenuse. Following the theorem, calculate \( 5^2 \) and \( 12^2 \) and then add these values: \( 5^2 + 12^2 = 25 + 144 = 169 \). Now, verify if \( 13^2 = 169 \) as well. Since the results match, we have correctly applied the Pythagorean theorem, confirming that these sides can form a right triangle.
For our specific problem, where the sides are 5, 12, and 13, we see that 13 is the longest side and is thus potentially the hypotenuse. Following the theorem, calculate \( 5^2 \) and \( 12^2 \) and then add these values: \( 5^2 + 12^2 = 25 + 144 = 169 \). Now, verify if \( 13^2 = 169 \) as well. Since the results match, we have correctly applied the Pythagorean theorem, confirming that these sides can form a right triangle.
Mathematical Reasoning
Mathematical reasoning is crucial for solving problems effectively and verifying the correctness of their solutions. This involves logical thinking and the ability to make reasoned judgments. In the context of verifying whether a set of lengths can form a right triangle, reasoning dictates that a correlation between the given lengths must be established following mathematical theorems or properties.
In the given exercise, mathematical reasoning is used to conclude that since the sum of the squares of the shorter sides equals the square of the longest side, as per the Pythagorean theorem (\( 5^2 + 12^2 = 13^2 \)), the lengths satisfy the condition needed for constructing a right triangle. If any of the initial assumptions were incorrect, or if the calculations did not result in equality, reasoning would guide us to conclude that a right triangle would not be possible with the provided lengths.
In the given exercise, mathematical reasoning is used to conclude that since the sum of the squares of the shorter sides equals the square of the longest side, as per the Pythagorean theorem (\( 5^2 + 12^2 = 13^2 \)), the lengths satisfy the condition needed for constructing a right triangle. If any of the initial assumptions were incorrect, or if the calculations did not result in equality, reasoning would guide us to conclude that a right triangle would not be possible with the provided lengths.
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