swerve wrote:In triangle ABC, side AB is 5 inches and side BC is 7 inches. What is the area of the triangle ABC?

1) Angle ABC is 90 degrees.
2) Triangle ABC is a right triangle.

The OA is A

Source: Manhattan Prep

Let's take each statement one by one.

1) Angle ABC is 90 degrees.

So, we have /_ABC = 90º, AB = 5 and BC = 7; thus, AB and BC are the legs of ∆ABC. Thus, the area of the triangle ABC = 1/2*5*7 = 17.5. Sufficient.

2) Triangle ABC is a right triangle.

Since we do not know which of the three angles /_A, /_B and /_C is 90º, we cannot have the unique value of the area of the triangle ABC.

Case 1: Say /_B = 90º, then as discussed in Statement 1, the area of the triangle ABC = 17.5.
Case 2: Say /_A = 90º, then BC = hypotenuse. Thus, AC = height = √(7^2 - 5^2) = 2√3. Thus, the area of the triangle ABC = 1/2*2√3*5 = 5√3.

No unique answer. Insufficient.

The correct answer: A

Hope this helps!

-Jay
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