Which three lengths could be used to form a right triangle?
Understand the Problem
The question is asking which combination of three lengths can form a right triangle. This involves applying the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
Answer
The lengths that could be used to form a right triangle are 51, 68, and 85.
Answer for screen readers
The lengths that could be used to form a right triangle are 51, 68, and 85.
Steps to Solve
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Apply the Pythagorean Theorem To check which set of lengths can form a right triangle, we use the Pythagorean theorem: $$ c^2 = a^2 + b^2 $$ where $c$ is the length of the hypotenuse (the longest side), and $a$ and $b$ are the other two sides.
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Check each option We will check each set of lengths given in the options one by one.
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Option A: 19, 36, 57 Identify $c = 57$, $a = 19$, $b = 36$. Calculate: $$ 57^2 = 3249 $$ $$ 19^2 + 36^2 = 361 + 1296 = 1657 $$ Since $3249 \neq 1657$, this option does not form a right triangle.
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Option B: 7, 10, 14 Identify $c = 14$, $a = 7$, $b = 10$. Calculate: $$ 14^2 = 196 $$ $$ 7^2 + 10^2 = 49 + 100 = 149 $$ Since $196 \neq 149$, this option does not form a right triangle.
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Option C: 15, 15, 48 Identify $c = 48$, $a = 15$, $b = 15$. Calculate: $$ 48^2 = 2304 $$ $$ 15^2 + 15^2 = 225 + 225 = 450 $$ Since $2304 \neq 450$, this option does not form a right triangle.
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Option D: 51, 68, 85 Identify $c = 85$, $a = 51$, $b = 68$. Calculate: $$ 85^2 = 7225 $$ $$ 51^2 + 68^2 = 2601 + 4624 = 7225 $$ Since $7225 = 7225$, this option forms a right triangle.
- Final conclusion After analyzing all options, we find that only option D forms a right triangle.
The lengths that could be used to form a right triangle are 51, 68, and 85.
More Information
The Pythagorean theorem is a fundamental principle in geometry that helps in determining whether a triangle is a right triangle based on the lengths of its sides. It is applicable in various fields, from architecture to physics.
Tips
- Failing to identify the longest side can lead to an incorrect application of the theorem.
- Mixing up the squares in calculations can result in wrong outcomes. It's helpful to carefully recheck arithmetic to avoid errors.
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