In the triangle above, DE is parallel to AC. What is the length of DE?

Hi All,

This DS question is built around the concept of similar triangles, which is a relatively rarity on Test Day (you probably will not see it, and if you do see it, it would likely be just once).

Similar triangles are triangles with the exact same angle measurements, but different side lengths. When dealing with similar triangles, the 'key' to solving for any of the side lengths, the area, the perimeter, etc., is to determine how any pair of equivalent sides relates to one another (e.g. if the 'first side' in the small triangle is exactly a third of the 'first side' in the larger triangle, then THAT relationship will hold true for the other 2 paris of sides).

This question is designed in such a way that you don't have to do any calculations to solve it, as long as you understand the 'rules' involved.

Here, since DE is parallel to AC, Angle A = Angle BDE and Angle C = Angle BED. Thus, we have similar triangles. We're asked for the length of DE.

Fact 1: AC = 14

This Fact is not enough to define the relationship or any other side lengths.
Fact 1 is INSUFFICIENT

Fact 2: BE = EC

This Fact tells us that BE is HALF of BC, which DOES define the relationship, but does NOT give us any values to work with.
Fact 2 is INSUFFICIENT

Combined,
Fact 1 gives us a value
Fact 2 defines the relationship.
Thus, we CAN determine the value of DE (it's half of AC)
Combined, SUFFICIENT

Final Answer:

GMAT assassins aren't born, they're made,
Rich

MANHATTAN GMAT OFFICIAL SOLUTION:

(1) INSUFFICIENT: Many elements in this triangle could vary; we don't even know the placement of B relative to AC, so the triangle itself might stretch. Even for a fixed triangle, we see that DE could slide up or down, so various lengths are possible for DE.


(2) INSUFFICIENT: We don't know the lengths of any sides of the triangle. The side that most affects the length of DE is AC, so we'll stretch that side. As we see, stretching the triangle out to the right stretches DE.


(1) AND (2) SUFFICIENT: AC must be 14, and DE must be parallel to AC and halfway between AC and B, in order to maintain BE = EC. Even though vertex B is free to move, DE will always be the average of the width of the triangle at AC (14) and the width at B (0). Thus, DE must be 7, no matter how the picture shifts.


The correct answer is C.

I have a small question here. If the question stem would have mentioned that point D is the midpoint of side AB and E is the midpoint of BC, then statement 1 alone would be sufficient since from the midpoint theorem DE||AC and therefore DE=1/2(AC). Am I correct??

I have a small question here. If the question stem would have mentioned that point D is the midpoint of side AB and E is the midpoint of BC, then statement 1 alone would be sufficient since from the midpoint theorem DE||AC and therefore DE=1/2(AC). Am I correct??

Statement 1

We not know anything about the ratio of the sides- is this an equilateral triangle?

Insuff

Statement 2

We do not know anything about the actual lengths of the sides

Insuff

Statement 1 & 2

If we know the AC, then we know the length of BE because and EC because they must be equal in order for DE to be the midline between AC and B

C

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