In the figure shown, if the measure of angle BAD is 60 degre

Question

screen_shot_2011_01_06_at_12.27.16_pm.png In the figure shown, if the measure of angle BAD is 60 degrees, ABCD is a parallelogram, and line BC bisects line AE, what is the length of side EC? (1) The length of DC is 12. (2) The length of BC is 12. Image file https://s3.amazonaws.com/production.gro ... .16_pm.png Grockit

mkdureja · Answer

https://s20.postimg.org/fws7lafjx/image.jpg

Statement 1:
DC is given.
Since there is no restriction on BC and AD, you can extend the points C and D to C' and D' without violating any given condition.
EC is now different to EC', so we can't answer the question on basis of this.

Statement 2:
BC is given.
Extend BA and CD to BA' and CD' respectively. Since length of AB=BE, therefore, A'B = BE'.
So, in this case also EC and E'C are different, so can't answer.

Combining:
This fixes the parallelogram, we can't alter AB or BC, so BE is also fixed(equal to AB), so EC is also fixed, as we can't alter the angle either.
Can answer when we combine the statements.

vigrah · Answer

angle BAD is 60. 
so angle ABC is 120 and angle CBE is 60
statement 1:
it tells us BE =12. with this information we cannot find the value of EC
Satement 2: 
it tells us BC= 12. with this info also we cannot find the value od EC.
when we combine both statements
we know BE=12 and BC=12 
angle at CBE is 60
since BE and BC are same angle BCE= angle BEC
now we know that BCE+BEC=120 and BEC=BCE hence triangle BEC is equalteral triangle
so length of EC is 12
Answer is C

WholeLottaLove · Answer

Why wouldn't BEC be similar to BCD?

WholeLottaLove · Answer

I think there is a problem with the site...every time I try to edit my post I get brought to the control panel. Furthermore, about 1/2 of the time I click on a post, I get brought to the control panel.

WholeLottaLove · Answer

(I cannot edit my last post because it keeps bringing me to the control panel so I will remedy my mistake here)

I assumed that Parallelogram ABCD would have equal interior angles if I were to draw a transversal from B to D. This is false. If BC and AD were compacted or expanded, they would alter the angle at which line BD passes through the parallelogram. This makes the rest of my proof false, as I assumed triangle BEC would be similar to BCD which I said was a equilateral triangle.

1.) We know that BA = BE. We also know that CD = BA. Furthermore we know that BC = AD and that angle BCD = angle BAD = 60. However, depending on the length of BC and AD, line EC could take on a myriad of lengths. There is no way to tell the length of EC from this. Insufficient.

2.) BC = 12. This gives us the length of BC and consequently, AD. However, we know nothing about the length of. BE. Depending on it's height. EC could take on a myriad of lengths. Insufficient.

1+2) This confirms my assumption before that ABCD is in fact a Rhombus. If this is the case, we know the length of BE and BC. We also know that angle EBC = 60. Even if we couldn't figure out the exact length we would know that there would only be one possible outcome for EC because the triangle has two defined lengths plus a known angle in between them (i.e. the triangle is locked into a single position.) Sufficient.

C

In the figure shown, if the measure of angle BAD is 60 degre

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