Which three lengths could be used to form a right triangle?
Understand the Problem
The question is asking which set of three lengths can form a right triangle based on the Pythagorean theorem. To solve it, we will check each option and apply the theorem to see if the square of the largest length equals the sum of the squares of the other two lengths.
Answer
The lengths that can form a right triangle are $51$, $68$, and $95$.
Answer for screen readers
The set of lengths that can form a right triangle is D: 51, 68, 95.
Steps to Solve
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Identify the largest side For each group of lengths, determine which length is the largest, as it will be considered the hypotenuse of the triangle.
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Apply the Pythagorean theorem Use the Pythagorean theorem: $c^2 = a^2 + b^2$, where $c$ is the largest side (hypotenuse) and $a$ and $b$ are the other two sides.
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Calculate for each option
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For Option A: Lengths 19, 36, and 57:
- Check if $57^2 = 19^2 + 36^2$
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For Option B: Lengths 7, 10, and 14:
- Check if $14^2 = 7^2 + 10^2$
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For Option C: Lengths 15, 15, and 48:
- Check if $48^2 = 15^2 + 15^2$
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For Option D: Lengths 51, 68, and 95:
- Check if $95^2 = 51^2 + 68^2$
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Evaluate each equation Compute the squares for each side and see if the equality holds true.
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Conclude which option forms a right triangle Determine which of the groups satisfies the Pythagorean theorem.
The set of lengths that can form a right triangle is D: 51, 68, 95.
More Information
The Pythagorean theorem is essential for identifying right triangles in geometry. A right triangle is characterized by one angle being exactly 90 degrees, which necessitates that the square of the length of the hypotenuse equals the sum of the squares of the other two sides.
Tips
- Forgetting to square the lengths: Always ensure that you square each length before adding or comparing.
- Not identifying the right longest side: Misidentifying the hypotenuse can lead to incorrect calculations.
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