Triangle ABC

[JackyL793]

There is a data sufficiency problem that I had encountered that doesn't seem correct. The problem provides us with a diagram of a (seemingly) equilateral triangle with a line drawn down the centre, perpendicular to the base. This is confirmed to be perpendicular as it is indicated with a right angle.

The question stem then provides us the height (6 + root 3) of the triangle (perpendicular line) and asks if we can solve the area of the triangle.

1) the Angle of the bottom left corner is 60 degrees
2) the Length of the opposite side of the triangle is 12

The solution tells us that since the triangle must be a 30-60-90 triangle, we would be able to solve for half of the length of the base using either 1) or 2), but that we would not be able to determine the other half without the other piece of information.

If we know that the height is perpendicular to the base, and connects to the tip of the triangle, wouldn't that split the base into 2, thus allowing us to solve for the area?

The only solution to this issue i can think of would be the fact that the diagram never explicitly states that the line connects to the tip of the triangle, but if that is the case, how would it ever convey that information?

Not too sure how to upload a picture, hope this question makes sense.

[Whit Garner]

Great question!

Statement 1 gives information that only helps with the left side of the triangle. Given the 60 degree measure of angle B, you know that A, B, and D have a fixed ratio of the sides (it will be a 30-60-90 triangle). However, while it looks like D cuts the line BC in half, geo isn't always drawn to scale (you can't trust your eyes). The points A, B, and D could stay where they are, but C could slide out 200 miles to the right. Or it could slide to the left until it sits right beside point D. NOT sufficient.

Statement 2 is the same issue, but this time for the left triangle. On the right you know that for DA to be a set length, and AC to be a set length, you can solve for DC. However, now you don't know anything about what is happening with the triangle on the left. The angle at B can be anything, so you could slide that point in or out along the line as far as you'd like. NOT sufficient.

Putting the 2 together gives you everything you need to determine all of the missing side lengths!

Hope this helps!