In the figure above, is the area…
Let’s start by sketching out the figure and labeling the sides.
We need to determine whether the area of triangular region ABC equal is equal to the area of triangular region DBA. Let’s label the sides with some variables. Let u = AC, v = BC, x = AD, y = AB and z = BD. We use the formula for the area of a triangle: A = (1/2)(b x h), or (b x h)/2. Thus, we can ask:
Is uv/2 = xy/2 ?
Is uv = xy ?
Statement One Alone:
AC^2 = 2(AD)^2
We can rewrite statement one as:
u^2 = 2x^2
√(u^2) = √(2x^2)
u = x√2
Since u = x√2, we can substitute x√2 for u in the question “Is uv = xy?”
Is (x√2)v = xy ?
Divide both sides by x:
Is v√2 = y ?
We see that we cannot answer the question. Statement one alone is not sufficient to answer the question. We can eliminate answer choices A and D.
Statement Two Alone:
Triangle ABC is isosceles.
With the information in statement two we know triangle ABC is an isosceles right triangle (i.e., a 45-45-90 triangle). Thus, the length of its hypotenuse is √2 times the length of one of its legs.
We see that y = u√2 or y = v√2. However, this is not enough information to determine whether the area of triangle ABC is equal to the area of triangle DBA because we don’t know anything about x. Statement two alone is not sufficient. We can eliminate answer choice B.
Statements One and Two Together:
From statement one we have rewritten the question to ask:
Is v√2 = y ?
From statement two we determine that y = v√2, so together the statements answer the question.