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PS_Possible Isosceles

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vittovangind Junior | Next Rank: 30 Posts
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PS_Possible Isosceles

Post Tue Jan 28, 2014 10:59 pm
Is triangle ABC an isosceles triangle?

(1) AD = DC

(2) The area of triangle ABD equals the area of triangle BDC
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Post Wed Jan 29, 2014 7:36 am
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

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Last edited by Brent@GMATPrepNow on Thu Apr 19, 2018 2:15 pm; edited 1 time in total

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Amrabdelnaby Master | Next Rank: 500 Posts Default Avatar
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Post Wed Nov 25, 2015 11:22 am
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

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Post Wed Nov 25, 2015 11:30 am
Amrabdelnaby wrote:
If this triangle was an equilateral triangle, statement one and two would have also been true.
So why are they sufficient?
Great question!

From the Official Guide for GMAT Review:
An isosceles triangle has at least 2 sides of the same length.

So, an equilateral triangle is also an isosceles triangle. We can say that an equilateral triangle is a special kind of isosceles triangle.

Similarly, a square is is a special kind of rectangle.

Cheers,
Brent

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Post Wed Nov 25, 2015 1:54 pm
Hi Amrabdelnaby,

You stated something in your last post that might need some clarification:

The two Facts/Statements are always TRUE - they represent new pieces of data that you can use to attempt to answer the given question (this is why they're called Data Sufficiency questions - you're looking to prove whether the data is sufficient to definitively answer whatever question is asked).

The two Facts can sometimes be REDUNDANT though (meaning that they offer data that you already had from the prompt, or that they offer the same data as each other).

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Max@Math Revolution Legendary Member
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Post Thu Nov 26, 2015 7:24 am
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.

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Post Fri Apr 01, 2016 11:40 am
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

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Post Fri Apr 01, 2016 12:21 pm
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

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Post Fri Apr 01, 2016 12:22 pm
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

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Marty Murray Legendary Member
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Post Fri Apr 01, 2016 10:36 pm
Rohit_Prakash88@yahoo.in 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.

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Post Sat Apr 02, 2016 12:37 am
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

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Post Sat Apr 02, 2016 6:36 am
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

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Max@Math Revolution Legendary Member
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Post Sun Apr 03, 2016 11:48 pm
Rohit_Prakash88@yahoo.in 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).

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Only $79 for 3 month Online Course
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