Question

What is the area of triangle ABC(see figure above)?
(1) DC=20
(2) AC= 8

• A. Data in statement I alone is sufficient.
• B. Data in statement II alone is sufficient.
• C. Data in both statements together is sufficient.
• D. Data in both statements together is not sufficient
• E. Either of the 2 statements is sufficient.

Answer 1

The correct option is E Either of the 2 statements is sufficient.

First, note that this is a Value Data Sufficiency question.

A big mistake in this problem would be to plunge into the statements without fully analyzing and exploiting the figure. You've got two right triangles that share a 90 span on either side of point C. What's going on here?

As it turns out, these triangles are similar.

Any time two triangles each have a right angle and also share an additional right angle (or, in this case, the ${90}^o$ span at point C), they will be similar. But if you didn't know that, you could easily uncover that fact by labeling any angle as x and labeling the others in terms of x:

Once you determine that both triangles have the angles ${90}^o,x$ and $90-x$, you may wish to redraw one or both of them in order to get them facing in the same direction.

Now, decide exactly what the question is asking. You need the area of triangle ABC. In order to get that, you need the base and height of that triangle.

Since the two triangles are right triangles, if you had any two sides of triangle ABC, you could get the third. Because the two triangles are similar, you could use any two sides of triangle CDE (note that you already have that side DE = 16), as well as the ratio of one triangle's size to the other, to get the third side of CDE as well as all three sides of ABC.

Thus, the rephrased question is, "What are any two sides of ABC, or what is any additional side of CDE plus the ratio of the size of each triangle to the other?"

(1) SUFFICIENT: Side DC equals 20. Use the 20 and the 16 to get, via the Pythagorean theorem, that side CE equals 12 (or simply recognize that you have a multiple of a 3-4-5 triangle). If CE equals 12 then AC equals 8. Thus, you have all three sides of CDE, plus the ratio of one triangle to the other (side AC, which equals 8, matches up with side DE, which equals 16; thus the smaller triangle is one-half the side of the larger).

Note that it is totally unnecessary to calculate further (once you have correctly rephrased the question, don't waste time doing more than is needed to answer the rephrase!), but if you are curious:

(2) SUFFICIENT: Side AC equals 8. Note that this gives you the same information as Statement 1. If AC equals 8, then CE equals 12 and you can calculate all three sides of CDE. Once you know that side AC equals 8 and that AC matches up with DE, which is equal to 16, you can know all three sides of ABC, as above.

The correct answer is (D).