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
