DAT QotD Solving for Hypotenuse Given Area and Perimeter

[acemason]

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I feel dumb asking this but I haven't done this sort of math since middle/elementary school so I'd appreciate some help. I got the right answer but I feel like the way I'm solving it takes too long. Please look over my method and let me know if there's a faster way/shortcut to solve this problem.

Today's DAT Question of the Day is:

If the area and perimeter of a right triangle are 30 cm2 and 30 cm, respectively, what is the length of the hypotenuse (the side opposite the right angle)?


So I'm solving this by first setting up two equations: Area: (1/2)(a)(b) = 30 Perimeter: a + b + c = 30 along with a^2 + b^2 = c^2.

First, I rewrote the Perimeter equation as: a + b = 30 - c

Then, I squared both sides of the equation and expanded: a^2 + b^2 + 2ab = 900 + c^2 - 60c

I then combined this with the a^2 + b^2 = c^2 equation and simplified it to: 2ab = 900 - 60c

From the Area equation we know (a)(b) = 60 so plugging this into the last equation gives: 120 = 900 - 60c

120 = 900 - 60c
60c = 780
c = 13 i.e. length of the hypotenuse.

 

[FeralisExtremum]

It's a fairly long problem to solve, but the way you did it was correct and as far as I'm aware there isn't any shorter method (aside from maybe recognizing that a 5-12-13 triangle happens to satisfy both equations).

 

[ns718]

That looks way too hard and time consuming for the DAT. How do they expect us to do that?

 

[matardug]

I saw nothing remotely close to a question that time consuming on either of my DATs, bad question IMO.

 

[Ari Rezaei]

The point of this question is to encourage you to find the shortcut. If you see a triangle on the DAT, try the Pythagorean Triples or special right triangles from the start. You might save a lot of time than doing it the long way (and sometimes seemingly impossible way).