Geometry Question on MGMAT Question Bank

[abramson]

Hello, This is in reference to Q.20 (Unknown Leg) on the MGMAT Geometry Question Bank.

I understand that statement (1) is sufficient. For statement (2), however, consider the following:

In the diagram given, drop a line from B, perpendicular to AC. Mark this point on AC as D. BD is now the height of the triangle. Now, since we know area of triangle is 30, and we know (1/2)(base)(height) is area, (1/2)(base) in this case would be (1/2)(side AC), which is 6.5, and height is BD. From here we get BD = (30/6.5). Using this height now, given BD is perpendicular to AC, in triangle BAD, we can now find the length of AD using Pythagorean theorem, since we know AB and BD. Subtract this new length of AD from AC (given = 13) to find DC. Now we know DC and DB, and angle BDC = 90, so we have another right triangle. Using Pythagorean theorem, we can now find length of BC.

Using this argument, I arrived at (D) for my answer. Can anyone confirm that this reasoning is correct/flawed? and why?

Thanks!

[Harish Dorai]

I second this approach and the reasoning is correct. The area of a triangle can also be calculated using the formula you mentioned. With the value of BD we can find the length of the 3rd side BC. Great approach!

[anadi]

Consider a situation where angle BAC > 90' and we have 2 possible values for line segment BC.