Let the two legs have lengths \(2x\) and \(5x\). Doing so, ensures that the ratio of the lengths of the two legs is \(2\) to \(5\)


The area of triangle ABC is 20

Area of a triangle \(= \frac{(base)(height)}{2}\)

Since we're told the area is \(20\), we can write \(\frac{(5x)(2x)}{2}=20\)

Simplify: \(\frac{10x^2}{2}=20\)
Multiply both sides of the equation by \(2\) to get: \(10x^2 = 40\)

Divide both sides by \(10\) to get: \(x^2 = 4\)

Solve: \(x = 2\) OR \(x = -2\)
Since the side lengths can't be negative, we can be certain that \(x = 2\)

Since the two legs have lengths \(2x\) and \(5x\), can now say the lengths of the legs are \(4\) and \(10\)

What is the length of the hypotenuse?
Let h = length of the hypotenuse

Applying the Pythagorean Theorem, we get: 4² + 10² = h²
Simplify: 16 + 100 = h²
Simplify: 116 = h²
So, h = √116 = (√4)(√29) = 2√29

Answer: E

Cheers,
Brent
_________________

Ok, this one almost got me with a ratio AND geometry. Like come on! But, I figured it out.

So, we know that the area is 20. And we know the legs are 2:5.

20 = 1/2(bh)
Solving for bh, we get:
bh = 40

I solved for bh so I could see which 2:5 ratio would multiply to 40. Easy, 10*4 = 40! And 4:10 reduces to 2:5, so it fits!

Now that I know my legs are 4 and 10, I can use the pythagorean theorem to solve for the hypotenuse (or c in the equation).

a^2 + b^2 = c^2
4^2 + 10^2 = c^2
16 + 100 = c^2
116 = c^2

From here, I went straight to the answers to see how they reduced this. I knew there had to be a square root in the answer because 116 isn't a perfect square and isn't a multiple of perfect squares.

I looked at answer C, but that couldn't work because I'd have to have sqrt(5)*sqrt(16) to get to 4sqrt(5). 16*5 does not equal 116, so that's out.

Then I looked at Answer E. To get there I'd need to see if sqrt(4*29) = sqrt(116), because sqrt(4) would have reduced to 2, and left me with 2sqrt(29).
29*4 = 116, so this one is the answer.

(If I knew how to use the formatting, this would be way easier to read but I don't...so I hope this makes sense!)

Answer is E

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