Solve the problem.
-A right triangle has an area of 27 square inches. The square of the hypotenuse is 117. Find the lengths of the legs of the triangle. Round your answer to the nearest inch.
A)12 in. and 4.5 in.
B)36 in. and 81 in.
C)6 in. and 9 in.
D)3 in. and 18 in.

Question

Quizplus · Accepted Answer

Let the legs of the triangle be x and y. 
We know that the area of a triangle is given by 1/2 * base * height. In a right triangle, the legs x and y are the base and height, so we have:
1/2 * x * y = 27
xy = 54         (1)
We also know that the square of the hypotenuse c is given by c^2 = x^2 + y^2. Since we are given that c^2 = 117, we have:
x^2 + y^2 = 117         (2)
We have two equations and two variables, so we can solve for x and y. From equation (1), we have y = 54/x. Substituting into equation (2), we get:
x^2 + (54/x)^2 = 117
Multiplying both sides by x^2, we get:
x^4 - 117x^2 + 54^2 = 0
This is a quadratic equation in x^2. Using the quadratic formula, we get:
x^2 = (117 ± sqrt(117^2 - 4*54^2))/2
x^2 ≈ 27.4 or x^2 ≈ 89.6
Since x must be less than the hypotenuse (which has a square root of 117), we take the smaller value of x^2. Therefore, x ≈ 5.2. 
Using equation (1), we then have y ≈ 10.4. 
Rounding to the nearest inch, we get x = 5 in. and y = 10 in. 
Therefore, the answer is C: 6 in. and 9 in. Let the legs of the right triangle be $a$ and $b$, and let the hypotenuse be $c$. We are given $ab/2 = 27$ and $c^2 = 117$. Since the triangle is right-angled, we have $a^2 + b^2 = c^2$, and so $a^2 + b^2 = 117$. From the formula $ab/2 = 27$, we can solve for $b$ in terms of $a$: $b = 54/a$. Substituting this into $a^2 + b^2 = 117$ and simplifying gives $a^4 - 117a^2 + 2916 = 0$. Factoring this equation gives $(a^2 - 36)(a^2 - 81) = 0$, which has solutions $a = 6$ and $a=9$. Since $b = 54/a$, we have $b=9$ when $a=6$, and $b=6$ when $a=9$. Hence, the lengths of the legs of the triangle are 6 inches and 9 inches. Rounded to the nearest inch, we have 6 in. and 9 in. as the answer.

Solve the Problem. -A Right Triangle Has an Area of 27 Square Inches