Geometry Question Bank #20 : Unknown leg.

[priti]

Data Sufficiency problem:

Given a triange ABC, and length(AB) = 5, length(AC) = 13,
What is length (BC) ?

A) Angle ABC is 90 degrees
B) Area os ABC is 30.

A) is sufficient, as we can then use pyth. thm.
B) is also sufficient, as we can use a base and area to find height. Then use height and one of the sides to find out how the height divides the base. Once we know how the base is divided, we can then find the other side.

So, the answer is D

However, the suggested answer is A. The test key doesn't clearly explain why B is not sufficient.

[Guest]

1. sufficient since we know that the triangle is a right triangle and therefore this is 5-12-13 triangle.

so AD

2. we do not know which leg is the base; it could be 5, 13,or the unknow base segment BC. If bc is the base then we don't know the height. insufficient

hope this helps.

Harris

[StaceyKoprince]

There are two different ways you could solve with statement 2.

So, any one of the three bases times the height associated with that base = the area.
eg, (1/2)(5)h = 30; h = 12 (for that base). Let's use this one.

I can draw two different triangles with this: one in which the angle between AB and AC is acute and one in which the angle between the two is obtuse. (Get a piece of paper and a pen and try it.)

So, for one of those, I can indeed figure out where the height "cuts" the base 5 and calculate the resulting third leg of the triangle. But that's only one of two ways I could draw this triangle... :)

[griffin.811]

So Im still not understanding the previous explanation for why (2) isn't sufficient to some extent, and like the OP, I too chose D for this one.

Is the underlying issue here that, despite being given the area of the triangle, the value of the height can still be different depending on which side we decide to use as the base?

I understand the concept of the height being outside the triangle but I would never have thought to do that. Was there something in the question that should signal that we should try an obtuse angle with the height outside the triangle?

After staring at this for a while now, the best I can come up with for why (2) is insufficient on its own is because without knowing that angle ABC is 90 degrees, even if we knew what the height was, we wouldn't be able to say for certain that the three triangles that would be created by the height line are similar, which would mean we can't use ratio analysis to solve BC.

Is this approach sound?

Thanks!

[RonPurewal]

I'm too lazy to type lots of words right now, so I'll give you coordinates instead.

Possibility 1:
A(0, 0)
B(5, 0)
C(5, 12)
This is the familiar 5-12-13 triangle, which you surely know and love.

Possibility 2:
A(0, 0)
B(5, 0)
C(-5, 12)
There you go. If you use AB as the base, the height is still 12 (an observation that's easier to see with coordinates).

[griffin.811]

Makes sense!

[RonPurewal]

Excellent.