In triangle ABC above, what is the length of side BC?

Why is AD=BD=BC? If AD is opposite to X, BD opposite to X as well and BC opposite to 2x?
Shouldn't BC be larger than both or equal to both TOGETHER?

Thanks for clarifying

Cheers!
J

Yes but they are sides of different triangles. Note that by the same logic, BD is opposite to 2x as well. The point is that it is opposite to x in one triangle (ABD) and opposite to 2x in another triangle (BDC).

BC will be equal to BD because they are both opposite 2x in triangle BDC.

AD will be equal to BD because they are both opposite angle x in triangle ABD.

So AD = BD = BC

Hey, great explaination, thanks.

Question: Is it the case that whenever angles are equal, their sides must equal?


Is there no exception where angle a \(=\) angle b but length a \(=/=\) length b?

Yes, the base angles of an isosceles triangle are always equal and vise-versa: if two angles in a triangle are equal then it's an isosceles triangle.

Why did you assume that ADC is a straight line. Because if it isint given specifically in the question then the entire logic fails. then angle BDA and BDC are not supplementary.

OFFICIAL GUIDE:

Problem Solving
Figures: All figures accompanying problem solving questions are intended to provide information useful in solving the problems. Figures are drawn as accurately as possible. Exceptions will be clearly noted. Lines shown as straight are straight, and lines that appear jagged are also straight. The positions of points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero. All figures lie in a plane unless otherwise indicated.

Data Sufficiency:
Figures:
• Figures conform to the information given in the question, but will not necessarily conform to the additional information given in statements (1) and (2).
• Lines shown as straight are straight, and lines that appear jagged are also straight.
• The positions of points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero.
• All figures lie in a plane unless otherwise indicated.

The lines that appear straight are straight. Also ABC is a triangle (given). So AC is a straight line. D is a point on AC and we have been given the measure of angle BDC as 2x. Hence there is no ambiguity here.

PROMPT ANALYSIS

The figure has angle BAD =x, angle BDC =2x, angle BCD = 2x.

Super set
The side length of BC could be any positive real number.

Translation
Since Angle BAD + Angle ABD = Angle BDC therefore angle ABD = x. Hence triangle ABD and triangle BDC are isosceles triangles. Hence AD = BD = BC.
In order to find the length of BC we need:
1# exact value of BC.
2# relation or property that will lead us to find the length of BC.

Statement analysis
St 1: AD = 6. Since AD = BD = BC, therefore BC = 6 ANSWER. Option b, c and e.
St 2: x =36. Since there is no idea about any side of the the figure, therefore it is insufficient.

Hence option A

I'm not getting the concept underlying this problem. Could anyone please clarify me that and refer to a particular topic from any book/blog/website to learn this. Thanks . Bunuel mikemcgarry VeritasPrepKarishma

Hey sadikabid27 ,

Let me explain you.

We need to find out the length of BC.

We already know that BD = BC because triangle BDC is isosceles. --(1)

Also, angle ABD + angle BAD = angle BDC ( Sum of interior opposite angles = Exterior angle.)

=> angle ABD = 2x - x = x.

=> Triangle BDA is also isosceles.

=> AD= BD --(2)

from (1) and (2), we can say

AD = BC.

Now option A is giving us AD, Hence the value will be same for BC. Hence, sufficient.

Option B isn't giving us any length. Hence, insufficient.

Does that make sense?

Thanks for your fast response abhimahna. Maybe, it's very silly of me but I'm not getting these steps
"Also, angle ABD + angle BAD = angle BDC ( Sum of interior opposite angles = Exterior angle.)

=> angle ABD = 2x - x = x.

=> Triangle BDA is also isosceles"

Hey sadikabid27 ,

You need to learn the properties of the triangles.

One of the properties says : The exterior angle of a triangle is always equal to the sum of the two opposite interior angles.

Check this image:



Does that make sense?

There are two Geometry concepts being tested here:

1. If two angles of a triangle are equal, sides opposite to them are equal too (isosceles triangle).
2. Measure of the exterior angle is the sum of interior opposite angles. Check this: https://mathbitsnotebook.com/Geometry/S ... Angle.html

Even if you don't know the theorem (point 2), you can easily derive it. The exterior angle BDC makes a straight angle with BDA so their sum will be 180.
BDC + BDA = 180
Also, DAB + ABD + BDA = 180 ( sum of angles of a triangle)

BDC + BDA = DAB + ABD + BDA
BDC = DAB + ABD

Target question: What is the length of side BC?

Statement 1: Line segment AD has length 6.
BEFORE we deal with statement 1, let's see what information we can add to the diagram.

For example, since ∆BDC has 2 equal angles (of 2x°), we know that side BD = side BC:


Next, since angles on a line add to 180°, and since ∠BDC = 2x°, we know that ∠ADB = (180 - 2x)°



Now focus on ∆BAD
Since angles in a triangle add to 180°, we know that ∠ABD = x°
ASIDE: Notice that x° + x° + (180 - 2x)° = 180°



Now that we know ∆BAD has two equal angles (x° and x°), we know that side AD = side BD

This means AD = BD = BC

Statement 1 tells us that AD = 6, which means BC = 6
The answer to the target question is side BC has length 6
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: x = 36
Notice that our diagram doesn't any lengths.
We can SHRINK or ENLARGE the diagram and the angles remain the same.
However the length of side BC changes.

Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

Since angle BDC=2X and angle BAC=X. It means point D has to be the centre of the circle which passes through points A, B and C. It makes the line segments AD, BD and CD the radius of the circle. So if AD has length 6, BD and CD have also the equal length of 6. Now, BD=CD=6 and angle BCD= angle BDC; means triangle BDC is an equilateral triangle. Hence BC=6.

Similar Question: https://gmatclub.com/forum/in-the-trian ... 40198.html

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