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If triangle ABC is right angled at vertex A

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Source: — Data Sufficiency |

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by Jay@ManhattanReview » Wed Mar 14, 2018 5:10 am
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

OA: C
Given: The triangle ABC is right angled at vertex A.

=> BC is hypotenuse, AB and AC are altitude and base

Area of triangle ABC = 1/2*AB*AC

Let's take each statement one by one.

(1) AB + AC = 8

There are many possible values of 1/2*AB*AC given AB + AC = 8. Insufficient.

(2) The length of the largest side of the triangle is 5√2.

The largest side of a triangle is the hypotenuse, which is BC

=> BC = 5√2

Can't get the value of 1/2*AB*AC. Insufficient.

(1) and (2) together

From Pythagoras theorem, we have BC^2 = AB^2 + AC^2

From Statement (1), we have AB + AC = 8 => AB = 8 - AC.

Plugging-in the value of AB in BC^2 = AB^2 + AC^2, we get

(5√2)^2 = (8 - AC)^2 + AC^2

50 = 8^2 - 2*8*AC + AC^2 + AC^2

50 = 64 -16AC + 2AC^2

2AC^2 - 16AC + 14 = 0

AC^2 - 8AC + 7 = 0

AC^2 - 7AC - AC + 7 = 0

AC(AC - 7) - 1(AC - 7) = 0

=> (AC - 1) (AC - 7) = 0

AC = 1 or 7

Case 1: AC = 1, then BC = 7 and the area of triangle ABC = 1/2*AB*AC = 1/2*1*7 = 3.5
Case 2: AC = 7, then BC = 1 and the area of triangle ABC = 1/2*AB*AC = 1/2*7*1 = 3.5; unique answer

The correct answer: C

Hope this helps!

-Jay
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by GMATGuruNY » Wed Mar 14, 2018 6:25 am
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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