In ∆ ABC above, AB = BC, and CD bisects angle C. If y = x/3, then z =

Question

https://gmatclub.com/forum/download/file.php?id=37011 In ∆ ABC above, AB = BC, and CD bisects angle C. If y = x/3, then z = (A) 40 (B) 60 (C) 64 (D) 72 (E) 80 2017-08-18_1027_001.png

Luckisnoexcuse · Answer

angle BAC = angle BCA = z
angle BCD = angle DCA = z/2
in triangle ADC
180 = x+z+z/2 = x+3z/2.....(i)
in triangle BDC
180 = 180-x+y+z/2
x-y-z/2=0...........................(ii)
in triangle ABC
180=z+z+y
180=2z+y..........................(iii)

we know y=x/3
x=3y
2y=z/2
4y=z

2z+y=180
2z+z/4=180
9z/4=180
z=80, y=20 and x=60
E

generis · Answer

ABCxyztriangle.png Derivation of figure's annotations: AB = BC: ∠ A = ∠ C CD bisects ∠ C: two angles formed by the bisector are equal ∠ A = ∠ C, and ∠ A = z . So ∠ C = z Angles at ∠ C formed by bisector = z/2 Isolate one variable: 1) Given: y = \frac{x}{3} . x = 3y x shows up in the figure only once; replace it with 3y in figure 2 ∠ BDC and 3y are on a straight line. ∠ BDC = (180 - 3y) 2) All three triangles' angles measures now are expressed in terms of y and z. Find one variable in terms of the other. ∆ ABC: 2z + y = 180 ∆ ADC: z + \frac{z}{2} + 3y = 180 ∆ BDC: y + (180 - 3y) + \frac{z}{2} = 180 3) Use ∆ BDC first: with 180 on both sides of the equation, RHS will = 0. That's a good start. y + (180 - 3y) + \frac{z}{2} = 180 y + 180 - 3y + \frac{z}{2} = 180 -2y + \frac{z}{2}= 0 \frac{z}{2} = 2y , z = 4y 4) Find y , where z = 4y . ∆ ABC is simplest for substitution 2z + y = 180 2(4y) + y = 180 9y = 180 y = 20 5) If y = 20 , from z = 4y , z = 80 Answer E

PranayBudime · Answer

Given that AB=BC,
Therefore Angle C = Angle A = z
Also, CD is the angular bisector for angle C, so Angle BCD = Angle ACD = z/2 
and y= x/3

As we know Angle ADC+ Angle BDC = 180 (Straight angle)
x+Angle BDC =180 
Angle BDC = 180-x

Now that we have all angles 
Solve for Z using sum of angles property in triangles ADC and BDC 
In ADC,  z+x+z/2 =180 
=>  x+3z/2 =180  --(1)

In BDC,  x/3+180-x+z/2=180 
=>  z/2 =2x/3  
=>  x =3z/4  --(2)
use value of x from (2) in (1)
 3z/4 + 3z/2 = 180 
Therefore z=80

Ans: E

shivam2506 · Answer

In ∆ ABC z+y+c=180 or 2z+y=180 as AB=BC so z=c (Iso. triangle). So we can write it as 12z+2x=1080.....(1)

In ∆ ADC, z+x+c/2 = 180 or 3z+2x=360......(2)

Solving 1 & 2 we get z=80, So IMO "E" is correct answer.

Jsound996 · Answer

For these type of questions, I like to derive the equations so I can isolate the variable of choice, which is Z
From the Information above, we know that AB = BC, which means that ∠BAC = ∠BCA
∠BAC = Z, then ∠BCA = Z

∠ABC +∠BAC +∠BCA =180, or Y+Z+Z=180  Equation 1

Now look at Triangle △ADC
∠ADC is equal to x, or 3y, and ∠DCA is equal to 0.5z because Line CD bisects the triangle, which means it divides ∠BCA by 2.

∠ADC+∠DAC+∠DCA = 180
3Y+Z+0.5Z=180  Equation 2

Now that we have 2 unique equations, we can isolate Z and solve it.
Z = 80

ScottTargetTestPrep · Answer

First, we see that angle CDB is (180 - x), so angle BCD is:

180 - y - (180 - x) = x - y

Since y = x/3, we can substitute this for y, obtaining x - x/3 = 2x/3 for angle CDB

Since angle C is bisected, by CD, we know that angle ACD is also 2x/3.

Thus, angle C is 4x/3, which is also equal to z (because triangle ABC is isosceles).

So, finally, we have:

4x/3 + 4x/3 + x/3 = 180

9x/3 = 180

3x = 180

x = 60

Recall that 4x/3 equals angle z; thus z = 4(60)/3 = 4*20 = 80.

Answer: E

dkilroy · Answer

I have not done geometry in a while, so this took me about 10 minutes, but I got the right answer thinking it through and remembering some basic concepts. IMPORTANT: It is helpful to use x=3y instead of y=x/3! Kudos to all the correct solutions!

CEdward · Answer

Let this be a lesson that one algebra mistake can ruin it all...Took a colossal 9 minutes for a very straightforward question. This is all algebra.

y + 2z = 180 (Since we know that AB = BC) --> x/y + 2z = 180

For the smaller triangle:

x + z + z/2 = 180 --> 2z + z + 2x = 360 --> 3z + 2z = 360 --> 2x = 360 - 3x --> x = 180 - 3z/2 --> x = 360 - 3z/2

y = x/3 = 1/3 x (360 - 3z/2) = 360 - 3z/2 (plug this into the equation above)

360 - 3z/6 + 2z = 180 --> z = 80

E.

ThatDudeKnows · Answer

The two previous responses in this thread mention taking 10 minutes and 9 minutes to solve this. Yikes!! If you find yourself knowing right out of the gate that your scratch paper is going to look like alphabet soup, maybe try a different way! How about rather than doing a bunch of algebra, we just Plug In The Answers (PITA)?

B looks easy to work with. Let's try that (see diagram). Since AB=BC, z = the two angles at C combined. Answer choice B makes z=60. The two angles at C are both 30. x is 90. The supplementary angle at x is 90. That makes y=60. Does that fit for y=x/3? 60=(90/3)? Nope. B is wrong.

That only took a few seconds, so at this point I'd probably just try one of the other answer choices, likely A or E. Either way, by the time we try E, it's going to add up to maybe around a minute. Sure beats 9 or 10 minutes!!

If you get good at PITA, you'd only need to test D once you've already tried B. D gets us closer to where we want to go, but still not quite there. So it would have to be E.

bumpbot · Answer

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In ∆ ABC above, AB = BC, and CD bisects angle C. If y = x/3, then z =

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