NandishSS wrote:If triangle ABC is right angled at vertex A, what is the area of triangle ABC?
(1) AB + AC = 8.
(2) The length of the largest side of the triangle is 5√2
Since the right angle is at vertex A, AB and AC constitute the legs -- and thus are the base and the height -- of the triangle.
Thus:
Area = (1/2)
(AB)(AC).
To calculate the area, we need to know the value of the product in blue.
Question stem, rephrased:
What is the value of (AB)(AC)?
Statement 1:
Case 1: AB=1 and AC=7, with the result that AB+AC = 1+7 = 8
In this case, (AB)(AC) = 1*7 = 7.
Case 2: AB=2 and AC=6, with the result that AB+AC = 2+6 = 8
In this case, (AB)(AC) = 2*6 = 12.
Since (AB)(AC) can be different values, INSUFFICIENT.
Statement 2:
Since hypotenuse = BC = 5√2, and AB² + AC²= BC², we get:
AB² + AC² = (5√2)²
AB² + AC² = 50.
Case 1: AB=1 and AC=7, with the result that AB² + AC² = 1² + 7 = 50
In this case, (AB)(AC) = 7.
Case 3: AB=5 and AC=5, with the result that AB² + AC² = 5² + 5² = 50
In this case, (AB)(AC) = 5*5 = 25.
Since (AB)(AC) can be different values, INSUFFICIENT.
Statements combined:
Of the cases above, only Case 1 -- in which the legs of the triangle are 1 and 7 -- satisfies both statements.
Implication:
To satisfy both statements, either AB=1 and AC=7 or AB=7 and AC=1.
Thus:
(AB)(AC) = 7.
SUFFICIENT.
The correct answer is
C.
An algebraic way to combine the two statements:
Squaring AB+AC = 8, we get:
(AB+AC)² = 8²
AB² + AC² + 2(AB)(AC) = 64.
Substituting AB² + AC² = 50 into the equation in red, we get:
50 + 2(AB)(AC) = 64
2(AB)(AC) = 14
(AB)(AC) = 7.
SUFFICIENT.
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