Does 5, 12, 13 make a right triangle?
Understand the Problem
The question is asking whether the numbers 5, 12, and 13 can represent the lengths of the sides of a right triangle. To determine this, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) equals the sum of the squares of the lengths of the other two sides.
Answer
Yes, the sides can form a right triangle.
Answer for screen readers
Yes, the sides 5, 12, and 13 can form a right triangle.
Steps to Solve
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Identify the sides of the triangle We'll first identify the longest side, which is the hypotenuse. In this case, we have the sides 5, 12, and 13. The longest side is 13.
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Apply the Pythagorean theorem The Pythagorean theorem states that for a right triangle: $$ c^2 = a^2 + b^2 $$ where $c$ is the length of the hypotenuse, and $a$ and $b$ are the lengths of the other two sides. Here, we substitute $c = 13$, $a = 5$, and $b = 12$.
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Calculate the squares of the lengths We will calculate the squares of 5, 12, and 13: $$ 13^2 = 169 $$ $$ 5^2 = 25 $$ $$ 12^2 = 144 $$
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Sum the squares of the shorter sides Now, we will sum the squares of the shorter sides: $$ 5^2 + 12^2 = 25 + 144 = 169 $$
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Compare the sums to the hypotenuse Finally, we compare $c^2$ to $a^2 + b^2$: $$ 169 = 169 $$ Since both sides are equal, this verifies that the triangle can indeed be a right triangle.
Yes, the sides 5, 12, and 13 can form a right triangle.
More Information
The numbers 5, 12, and 13 form one of the classic Pythagorean triples, meaning that they are integer solutions to the Pythagorean theorem.
Tips
- A common mistake is to misidentify the hypotenuse. Always confirm that the longest side is chosen as the hypotenuse.
- Another mistake is forgetting to square the lengths correctly, leading to incorrect comparisons.
