Number 20 from QUestion Bank: Geometry: Unknown Leg

What is the length of segment BC?

(1) Angle ABC is 90 degrees.

(2) The area of the triangle is 30.





Why would statement 2 be insufficient? Couldn't I draw a perpendicular line from point B to line AC (call it point D), which will give me the height? I could find the height using the equation Height * 13 (line AC) * 0.5 = 30. Thus height is 60/13.

I could then solve for length of AD, which will give me length of DC. I could then solve for line segment BC.

What's wrong with my logic?

Guest,

I think there is nothing wrong with your logic. I too think the answer is B.

The MG answer is A.

Is this an error? Shouldn't the answer be D?

I agree; this is an error.

You could even look at it another way: Draw a height from point C to hit side AB perpendicularly at some point we'll call point E. Using the formula for the area of a triangle (Area=0.5*base*height) would show that 30 = 0.5*5*EC, so EC must be 12. Since AEC is a right triangle, using Pythagorean theorem with EC=12 and AC=13, it must be that AE is 5--in other words, our created point E actually coincides with point B. Thus, we see from statement (2) that angle ABC is 90 degrees; this must be the common right triangle 5:12:13. Statement (1) told us directly that angle ABC is 90 degrees, so it is sufficient by the same logic.

The answer should be D.

I will make sure this gets corrected. Thanks!

Actually, statement 2 is not sufficient. You can "swing" the side AB (length 5) so that you have the same height dropped from point B down to AC (and therefore the same area), but you have a different BC length. Click on the following diagram to see why.

Christian- Can you explain your reasoning a bit more? I follow your diagram, but is there some rule that we should remember before we draw a bisector to find the height of a triangle? Thanks.

I stand corrected! Thanks Chris.

This is a great example of two traps:

a) Picture bias: The picture showed angle BAC < 90 degrees, so I made the subconcious assumption that it had to be.

b) Statement carryover: Statement (1) indicates that angle ABC is a right angle. When considering Statement (2), I should have been mindful of all the possible triangles allowed by Statement (2). Erroneously, my approach was more of a "check to see whether Statement (2) is equivalent to Statement (1)" approach.

A couple of practices to avoid such errors on Geometry DS questions:
a) Redraw the picture--taking care to draw it every way it can "look" according to the words and labels in the question.
b) When considering an individual statement, deliberately try to violate the other statement as you draw the picture. This will help you see whether the absence of the other statement's information makes a difference, and will prevent statement carryover.