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What is the perimeter of a certain right triangle?

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Source: — Data Sufficiency |
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by Vincen » Sun Apr 01, 2018 10:51 am
Hello.

Let's take a look.

Since we have a right triangle. we know that can say that the cathetus are "a" and "b" and the hypotenuse is $$c=\sqrt{a^2+b^2}.$$ Now, we have to find a+b+c.

(1) The hypotenuse's length is 5.

We have that $$c=5\ \ \ \Rightarrow\ \ c^2=25\ \Rightarrow\ \ \ a^2+b^2=25.$$ But this doesn't tell us anything about "a" and "b". Hence, this statement is NOT SUFFICIENT.

(2) The triangle's area is 4.5

The area of a right triangle is A=(1/2)*a*b. Hence, $$4.5=\frac{a\cdot b}{2}\ \ \Rightarrow\ \ a\cdot b=9.\ $$ This time again, we can't find the value of "a", "b" and "c" (NOT SUFFICIENT).

Finally, using both statements together we can have that: $$\left(a+b\right)^2=a^2+2ab+b^2=25+2\left(9\right)=25+18=43.$$ $$\Rightarrow\ \ \ a+b=\sqrt{43}$$ Therefore, the perimeter is $$P=\ a+b+c=\sqrt{43}+5\ .$$ SUFFICIENT.

This implies that the correct answer is the option C.

I hope it helps.
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by Jake@ThePrincetonReview » Fri Apr 06, 2018 2:12 pm
Here's why Statement 2 is not sufficient: You can prove that it's insufficient by using real numbers to arrive at two different answers.
In this case, Statement 2 tells you that the area, (1/2)(base)(height)= 4.5.
So that means that (base)(height) = 9

Well, the base and the height could be 3 and 3, making the perimeter of the triangle 6 + 3 sqrt2
BUT, the base and the height could also be 1 and 9, making the perimeter 10 + sqrt82

Hope that helps! This is a strategy I use often.
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