Problem Solving

BTGmoderatorDC wrote:In a right triangle, the longer leg is two more than three times the shorter leg, and the area of the triangle is 400. What is the length of the shorter leg?

A 16
B 20
C 25
D 32
E 40

OA A

Source: Magoosh

In a right triangle, the longest side is called the hypotenuse, and the two sides that meet to create a 90-degree angle are called the legs.
So, one leg will be the triangle's base, and the other leg will be the triangle's height.

Let x = length of the triangle's base (which, for this question, will be the SHORTER leg)

The longer leg is two more than three times the shorter leg
If x = length of shorter leg (aka the base), then...
3x + 2 = length of longer leg (aka the height)

The area of the triangle is 400. What is the length of the shorter leg?
Area of triangle = (base)(height)/2

We can write: (x)(3x + 2)/2 = 400
Multiply both sides by 2 to get: (x)(3x + 2) = 800
Expand left side: 3x² + 2x = 800
Set equal to zero to get: 3x² + 2x - 800 = 0
Hmmm, I don't want to solve this awful quadratic equation, so what can I do?

One option is to start plugging in each answer choice to see which one satisfies the equation, but that could take a while.

Another thing we might do is recognize that, if (x)(3x + 2) = 800 (an earlier equation we derived), and if x is an integer (which we know is true, given the answer choices), then it's quite likely that x and (3x+2) are both divisors of 800

So, for each answer choice (possible x-value), we might first test whether (3x+2) is a divisor of 800.
For example, if x = 20 (answer choice B) then 3x + 2 = 3(20) +2 = 62
Since 62 is NOT a divisor of 800, we might check another answer choice.

With this strategy, we can see that, if x = 16 (answer choice A) then 3x + 2 = 3(16) +2 = 50
Since 50 IS a divisor of 800, we might check to see whether this value of x satisfies the equation (x)(3x + 2) = 800
When x = 16, we get: (16)(50) = 800 PERFECT!!

Answer: A

Cheers,
Brent

Hi swerve,

We're told that in a right triangle, the longer leg is two more than three times the shorter leg, and the area of the triangle is 400. We're asked for the length of the SHORTER leg. This question can be solved in a couple of different ways - and it can be solved rather easily by TESTing THE ANSWERS.

To start, we know that the area of the triangle is 400, so we can set up the formula for Area:
A = (1/2)(Base)(Height)
400 = (1/2)(Base)(Height)
800 = (Base)(Height)

We're asked for the SHORTER side, so we should start with one of the smaller answers. Let's TEST Answer B first....

Answer B: 20
IF... the shorter side = 20

the longer leg is two more than three times the shorter leg....
longer side = 2 + (3)(20) = 62

With side lengths of 20 and 62, the product would be (20)(62) = 1240. This is clearly too big (it's supposed to be 800). Thus, we need the sides the be SMALLER. There's only one answer that fits....

Final Answer: A

GMAT assassins aren't born, they're made,
Rich