The area of the right triangle ABC: "Rightfully Triangular"

[thoughtrunner]

Some help with this problem please. It is from MGMAT 750 Advanced Quant homework and titled "rightfully triangular"

The area of the right triangle ABC is equal to 60 square inches. If the hypotenuse of the triangle is 17 inches long, what is the perimeter of the triangle, in feet?

10/3
5
6
23
40

The "correct" is A, 10/3, using complicated Algebra.

I don't see why 40 can't also be correct.

If hypotenuse of right triangle is 17, the other two sides can be 8 and 15 right?

8^2 + 15^2 = 64 + 225 = 289 = 17^2

8*15 = base*height = 120. Area = 1/2 (base*height) = 60, given by the problem.

Thus, the perimeter = 8 + 15 + 17 = 40.

I don't see why this is not an acceptable answer choice.

Below is how MGMAT solves it:
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Let's denote the hypotenuse of the triangle as c, and the two shorter sides as a and b. To find the perimeter of the triangle, we need to find the sum of the lengths of the two missing sides, i.e. (a + b).

Since the two shorter sides in the right triangle are perpendicular (one of them can be viewed as the base and the other as the height), the area of the triangle, 60 square inches, is equal to half the product of the two shorter sides: (ab)/2 = 60. Thus, ab = 120.

Also, in any right triangle, a2 + b2 = c2. Therefore, a2 + b2 = 172 = 289

To find the values of a and b, we would need to solve the following system of equations:

ab = 120
a2 + b2 = 289

However, given the large values in each equation and the presence of squares, solving this system for a and b would result in a time-consuming quadratic equation with large numerical values. Remember, however, that we are not interested in the individual values of a and b, but rather in their sum, (a + b). Focusing on the sum will allow us to avoid the unnecessary computational work.

Note that the left-hand side of the equation a2 + b2 = 289 closely resembles our template (a + b)2 = a2 + 2ab + b2. We are just missing 2ab to complete the template; also, note that the value of ab is known from the first equation, ab = 120. Therefore, we can add 2ab to both sides of the equation to complete the template and solve for (a + b) with minimal computational effort:

a2 + b2 = 289
a2 + b2 + 2ab = 289 + 2ab
(a + b)2 = 289 + 2(120)
(a + b)2 = 529 = 23 × 23

Since the values of a and b can only be positive, a + b = 23 inches. Therefore, we can find the perimeter: a + b + c = (a + b) + c = 23 + 17 = 40 inches. Finally, since the problem asks for the perimeter in inches, do not forget to convert inches into feet: 12 inches = 1 foot; 40 inches = 40/12 = 10/3 feet.

The correct answer is A.

[RonPurewal]

Thus, the perimeter = 8 + 15 + 17 = 40.

I don't see why this is not an acceptable answer choice.

that's in inches.
you want feet.

40 inches, divided by 12 = 40/12 = 10/3 feet.

sneaky, we are.

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by the way, the answer choices are NUMBERS. this GUARANTEES that there will be one, and only one, correct answer to the problem. therefore, if you find the 8-15-17 triangle (always a good idea to memorize these!), then you're DONE.

[thoughtrunner]

Ugh. Get the hard part right and overlook the two second conversion - story of my life.

At least my method of solving was correct.

Thanks for the response.

[Ben Ku]

Glad that helps!