Data Sufficiency

Is triangle ABC an isosceles triangle?

(1) AD = DC

(2) The area of triangle ABD equals the area of triangle BDC

If this triangle was an equilateral triangle, statement one and two would have also been true.
So why are they sufficient?

Brent@GMATPrepNow wrote:

vittovangind wrote:

Is triangle ABC an isosceles triangle?

(1) AD = DC
(2) The area of triangle ABD equals the area of triangle BDC

NOTE: I added labels v, w, x, y, and z to make it easier to reference the sides.

Target question: Is triangle ABC an isosceles triangle?

To show that triangle ABC is an isosceles triangle, we can either show that the triangle has two equal angles or two equal sides.

Statement 1: AD = DC (i.e., x = y)
There's a nice rule that says that the altitude of an isosceles triangle always bisects the opposite side. Since the altitude (z) bisects AC, we can conclude that triangle ABC is an isosceles triangle.
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Here's another way to show that statement 1 is sufficient.
Notice that the two smaller triangles that comprise triangle ABC are both RIGHT TRIANGLES.
The two triangles both SHARE SIDE z, and statement 1 tells us that side x = side y.
Since both smaller triangles are right triangles, we COULD use the Pythagorean Theorem to determine the hypotenuses (v and w) of the two triangles. MOREOVER, since we'd be plugging the SAME NUMBERS into the Pythagorean Theorem, the hypotenuses (v and w) will be the same length.
If v and w have the same length, then triangle ABC is an isosceles triangle.
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: area of triangle ABD = area of triangle BDC
Area of triangle = (1/2)(base)(height)
So, (1/2)(x)(z) = (1/2)(y)(z)
If we divide both sides by (1/2)(z), we see that x = y
Since statement 1 also told us that x = y, and since we already determined that statement 1 is sufficient, we can conclude that statement 2 is SUFFICIENT

Answer = D

Cheers,
Brent

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

Is triangle ABC an isosceles triangle?

(1) AD = DC

(2) The area of triangle ABD equals the area of triangle BDC

We can know everything about the triangle ABC if we know BD, AD, and CD, so there are 3 variables, but only 2 variables are given by the conditions, so there is high chance (D) will be the answer.
The congruent rule of a triangle includes knowing 2 sides and the angle in between, so condition 1 is sufficient
For condition 2, the lower side is same if the area is equal, so is sufficient as well
The answer becomes (D).

For cases where we need 3 more equations, such as original conditions with "3 variables", or "4 variables and 1 equation", or "5 variables and 2 equations", we have 1 equation each in both 1) and 2). Therefore, there is 80% chance that E is the answer (especially about 90% of 2 by 2 questions where there are more than 3 variables), while C has 15% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since E is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, C or D.

Hi

I did not get the following point "If this triangle was an equilateral triangle, statement one and two would have also been true. So why are they sufficient? "

please help me explain

Hello All,

Also as per the Variable Method discussed by the Math Revolution since there are 3 variable AB , AC and BC in the original condition but the statement 1 and statement 2 give one one equation each which means we have 3 variables and 2 equations . So in this case the ans options should have been D as per the concept discussed by Math Revolution

Hello All,

Also as per the Variable Method discussed by the Math Revolution since there are 3 variable AB , AC and BC in the original condition but the statement 1 and statement 2 give one one equation each which means we have 3 variables and 2 equations . So in this case the ans options should have been D as per the concept discussed by Math Revolution

[email protected] wrote:Hi

I did not get the following point "If this triangle was an equilateral triangle, statement one and two would have also been true. So why are they sufficient? "

please help me explain

The point of the question you quoted is that the statements work whether only the top two sides of the triangle are of equal length or all three sides are of equal length. So the triangle in the question could have either two sides of equal length or three sides of equal length.

The answer is that that situation does not create a problem because an equilateral triangle is considered a special case of an isosceles triangle. An isosceles triangle has AT LEAST two sides that have the same length. An equilateral triangle has three sides that have the same length. So an equilateral triangle fits the definition of an isosceles triangle.

Hi,

Please help clarifying
Is triangle ABC an isosceles triangle?

(1) AD = DC
(2) The area of triangle ABD equals the area of triangle BDC


As per the Variable Method discussed by the Math Revolution since there are 3 variable AB , AC and BC in the original condition but the statement 1 and statement 2 give one one equation each which means we have 3 variables and 2 equations . So in this case the ans options should have been E but the answer choice is D

Hi,

Please help clarifying
Is triangle ABC an isosceles triangle?

(1) AD = DC
(2) The area of triangle ABD equals the area of triangle BDC


As per the Variable Method discussed by the Math Revolution since there are 3 variable AB , AC and BC in the original condition but the statement 1 and statement 2 give one one equation each which means we have 3 variables and 2 equations . So in this case the ans options should have been E but the answer choice is D...


Please Illustrate how can we solve it using Variable approach method

[email protected] wrote:Hello All,

Also as per the Variable Method discussed by the Math Revolution since there are 3 variable AB , AC and BC in the original condition but the statement 1 and statement 2 give one one equation each which means we have 3 variables and 2 equations . So in this case the ans options should have been D as per the concept discussed by Math Revolution

-> According to the method of Variable Approach, 1)=2) is derived.(approximately 95% of the time is right)
That is, although there are a number of variables, the answer is most likely to be D if 1)=2).