Data sufficiency question- Manhattan question 700 level

[Jayashree Thota]

In the figure to the right, if point C is the center of the circle and DB = 7, what is the length of DE in triangle EDB?

(1) x = 60°

(2) DE || CA

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.[/img]

[Jay@ManhattanReview]

In the figure to the right, if point C is the center of the circle and DB = 7, what is the length of DE in triangle EDB?



(1) x = 60°

(2) DE || CA

Given: Point C is the center of the circle and DB = 7;
Question: What is the length of DE in triangle EDB?

Let's take each statement one by one.

(1) x = 60°

In ∆ABC, AC and BC are radii, thus the angles opposite them must be equal. Thus, /_ABC = /_BCA = xº = 60º. This means that /_ CAB = 180 - 60 - 60 = 60º. So, ∆ABC is an equilateral traingle.

However, the relationship between ∆ABC and ∆BDE is not established. Thus, we cannot get the value of DE. Insufficient.

(2) DE || CA

Again, in ∆ABC, AC and BC are radii, thus the angles opposite them must be equal. Thus, ∆ABC is an equilateral triangle.

Let's observe ∆ABC and ∆BDE under the light of DE || CA

/_BCA = /_BDE
/_BCA = /_BED
/_ABC is a common angle for both the traingles.

Thus, ∆ABC and ∆BDE are similar triangles.

Using the property of similar triangles, we have

BC/BD = AC/DE

3.5/7 = 3.5/DE

=> DE = 7. Sufficient.

The correct answer: B

Hope this helps!

-Jay
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[Jayashree Thota]

Hi Jay
Thanks for the explanation but i didn't understand how using statement II, we can prove ABC is an equilateral triangle.
My thought process is:
Since AC and BC are radii, angles CAB and ABC are equal
and angle ACB is x, as given in the question.
if x=60, then all angles are equal.
if x is not 60, for example 50,
then CAB+ABC=180-50=130,
so CAB=ABC=65
DE||CA, gives us information that both triangles are similar, but i don't find any clue to either find the angle for x, or conclude ABC is equilateral triangle.
For this reason, i selected C as the right answer
I didn't understand where my reasoning fell short.
It would be more helpful if u could stress at this point.
Thanks in advance!