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The hypotenuse of a right triangle is twice the length of one of its legs. The length of the other leg is six feet. Find the lengths of the three sides of the triangle

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Let's denote the lengths of the legs as \(a\) and \(b\), where \(b\) is the leg with a known length of 6 feet.

Given that the hypotenuse (\(c\)) is twice the length of one leg, we can express this relationship as:

\[ c = 2a \]

We also know the Pythagorean theorem for a right-angled triangle:

\[ c^2 = a^2 + b^2 \]

Substitute the expression for \(c\) from the first equation into the Pythagorean theorem:

\[ (2a)^2 = a^2 + 6^2 \]


\[ (2a)^2 = a^2 + 6^2 \]

Expand and simplify:

\[ 4a^2 = a^2 + 36 \]

Subtract \(a^2\) from both sides:

\[ 3a^2 = 36 \]

Divide by 3:

\[ a^2 = 12 \]

Take the square root of both sides. Since we're dealing with lengths, consider both the positive and negative square roots:

\[ a = \pm \sqrt{12} \]

Further simplify:

\[ a = \pm 2\sqrt{3} \]

So, the possible values for \(a\) are \(2\sqrt{3}\) and \(-2\sqrt{3}\).

Now, we know the lengths of the legs (\(a\) and \(b\)) and the hypotenuse (\(c\)):

\[ a = 2\sqrt{3} \, \text{feet} \]
\[ b = 6 \, \text{feet} \]
\[ c = 2a = 4\sqrt{3} \, \text{feet} \]

The lengths of the three sides of the triangle are \(2\sqrt{3}\) feet, \(6\) feet, and \(4\sqrt{3}\) feet.
User Josir
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