The hypotenuse of a right triangle is 10 cm…
We are given that the hypotenuse of a right triangle is 10cm. Let’s sketch a diagram:
Since we have a right triangle, we can use the Pythagorean Theorem.
a^2 + b^2 = c^2
x^2 + y^2 = 10^2
x^2 + y^2 = 100
We need to determine the perimeter of the triangle, which can be expressed as x + y + 10. Thus, if we can determine the value of x + y, we can determine the value of the perimeter.
Statement One Alone:
The area of the triangle is 25cm^2.
Using the information in statement one we can set up an equation for the area of the triangle.
area = (base x height)/2
area = xy/2
25 = xy/2
50 = xy
We now have two equations:
1) x^2 + y^2 = 100
2) 50 = xy (Note that if we multiply this equation by 2, we get 100 = 2xy)
We also know that we need to determine the value of x + y. To make x + y fit with our two equations we can raise the quantity (x + y) to the 2nd power to obtain:
(x + y)^2 = x^2 + y^2 + 2xy
We know from equations 1) and 2) that x^2 + y^2 = 100 and 2xy = 100.
By substitution, x^2 + y^2 + 2xy = 100 + 100 = 200
Because x^2 + y^2 + 2xy = (x + y)^2, we see that (x + y)^2 = 200
Finally, to determine the value of x + y, we must take the square root of both sides of the equation (x + y)^2 = 200. (We will consider only the positive square root because length is always a positive value.)
√(x + y)^2 = +/-√200
x + y = √100 x √2
x + y = 10√2
Thus, the perimeter of the triangle is 10 + 10√2. Statement one alone is sufficient to answer the question. We can eliminate answer choices B, C, and E.
Statement Two Alone:
The two legs of the triangle are equal lengths.
We can fill the information from statement two into our diagram.
We can now plug the above information into the Pythagorean Theorem.
a^2 + b^2 = c^2
x^2 + x^2 = 10^2
2x^2 = 100
x^2 = 50
√x^2 = √50 (Note that we consider only the positive square root because x is a length.)
x = √25 x √2
x = 5√2
Thus, the perimeter of the triangle is 5√2 + 5√2 + 10 = 10√2 + 10. Statement two alone is also sufficient to answer the question.