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# Which is greater b^2+2ad or (a+d)^2 - c^2

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Which is greater b^2+2ad or (a+d)^2 - c^2 [#permalink]  08 Feb 2018, 10:22
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58% (02:06) correct 41% (03:14) wrong based on 17 sessions
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#GREpracticequestion Which is greater b^2+2ad or (a+d)^2 - c^2.jpg [ 25.49 KiB | Viewed 510 times ]

 Quantity A Quantity B $$b^2+2ad$$ $$(a+d)^2 - c^2$$

A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.
[Reveal] Spoiler: OA
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I think the answer is C.

With the figure, I’d do this:

e^2= a^2+d^2 at the same time e^2=b^2+c^2 so, we can equal both equations:
a^2+d^2 = b^2+c^2
a^2+d^2 – c^2= b^2

Second, I'll compare the two options:
Quantity A

Quantity B

We can subtract “2ad” in both equations
Quantity A
b^2

Quantity B
a^2+d^2-c^2

We already know that “b^2= a^2+d^2 – c^2”

Please let me know if it is ok. It’s just my guess.
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Linamrs wrote:
I think the answer is C.

With the figure, I’d do this:

e^2= a^2+d^2 at the same time e^2=b^2+c^2 so, we can equal both equations:
a^2+d^2 = b^2+c^2
a^2+d^2 – c^2= b^2

Second, I'll compare the two options:
Quantity A

Quantity B

We can subtract “2ad” in both equations
Quantity A
b^2

Quantity B
a^2+d^2-c^2

We already know that “b^2= a^2+d^2 – c^2”

Please let me know if it is ok. It’s just my guess.

I dont think you can use pythagorean theorem, always look out for that. I used to fall for it many times.
Try without the theorem, your method is close to mine. I got B another friend D. Still not sure.
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Tricky problem. There are a few hurdles here. The first is that the quantities look pretty complicated. I'd try to simplify if possible. Notice that Quantity B contains (a + d)^2, which can be rewritten as a^2 + 2ad + d^2. Then we can subtract the 2ad from both sides, and shuffle the leftovers till we have b^2 + c^2 and a^2 + d^2. See the first picture for the algebra.

Next, we should recognize b and c as the sides of the rightmost triangle and a and d as the sides of the leftmost. They have the same opposite side, but the rightmost triangle has a 92° angle, while the leftmost has an 88° angle. If the angle were 90° then we would of course know that b^2 + c^2 would be the same as e^2. But what does it mean when the angle is either bigger or smaller than 90°? Easiest to just draw a picture and exaggerate the difference to see what it means. See the 2nd picture. Doing so shows, without any math, that the two sides on either side of the smaller angle must be larger, so their squares must also be larger. Thus, B is the answer.
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Intern
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SherpaPrep wrote:
Tricky problem. There are a few hurdles here. The first is that the quantities look pretty complicated. I'd try to simplify if possible. Notice that Quantity B contains (a + d)^2, which can be rewritten as a^2 + 2ad + d^2. Then we can subtract the 2ad from both sides, and shuffle the leftovers till we have b^2 + c^2 and a^2 + d^2. See the first picture for the algebra.

Next, we should recognize b and c as the sides of the rightmost triangle and a and d as the sides of the leftmost. They have the same opposite side, but the rightmost triangle has a 92° angle, while the leftmost has an 88° angle. If the angle were 90° then we would of course know that b^2 + c^2 would be the same as e^2. But what does it mean when the angle is either bigger or smaller than 90°? Easiest to just draw a picture and exaggerate the difference to see what it means. See the 2nd picture. Doing so shows, without any math, that the two sides on either side of the smaller angle must be larger, so their squares must also be larger. Thus, B is the answer.

Thanks I reached a similar conclusion but Im still not convinced. Can we have another method?

This is a friend method. I am more convinced of his method but not 100% sure.
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Intern
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The problem with the above method is that we cant really tell from the manipulation.

Look

- decrease + increase 0 no change, these are the subscripts meaning.
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Expert's post

Even though the discussion is quite interesting and useful per se, it is unuseful for the other students.
Who is trying to search for explanations will not find them due to the screenshots.

Whenever you do wanna a discussion unfold, is a good way to write all about as text.

The same is true when you post a new one question: the rules of the board forbid the use of screenshots.

Please refer to this https://greprepclub.com/forum/rules-for ... -1083.html
and this https://greprepclub.com/forum/qq-how-to ... -2357.html

Next time a similar topic will be locked.

regards
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@Carcass-

@trunks14-
Here's a simple way to imagine it: for any right triangle, the two legs squared will equal the hypotenuse squared. But what if you kept the two legs the same length and squeezed down the hypotenuse to something smaller? Then we know that the two legs squared will now be larger than the new hypotenuse squared, and we also know that the angle must now be less than 90°.

Similarly, if we widen the two legs, the hypotenuse would now have to be larger, but the angle now exceeds 90°.

This is actually a general rule:
a^2 + b^2 < c^2 when the angle is less than 90°
and
a^2 + b^2 > c^2 when the angle is greater than 90°

So from these thought experiments we know that in the original question the two sides of the triangle squared on either side of the 88° angle must be larger than the two sides of the triangle squared on either side of the 92° angle. So again the answer is B.

A picture would've been much easier and less wordy but that's not allowed, haha.
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Expert's post
No need to say sorry

It is just a matter to keep the board efficient.

regards
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SherpaPrep wrote:
@Carcass-

@trunks14-
Here's a simple way to imagine it: for any right triangle, the two legs squared will equal the hypotenuse squared. But what if you kept the two legs the same length and squeezed down the hypotenuse to something smaller? Then we know that the two legs squared will now be larger than the new hypotenuse squared, and we also know that the angle must now be less than 90°.

Similarly, if we widen the two legs, the hypotenuse would now have to be larger, but the angle now exceeds 90°.

This is actually a general rule:
a^2 + b^2 < c^2 when the angle is less than 90°
and
a^2 + b^2 > c^2 when the angle is greater than 90°

So from these thought experiments we know that in the original question the two sides of the triangle squared on either side of the 88° angle must be larger than the two sides of the triangle squared on either side of the 92° angle. So again the answer is B.

A picture would've been much easier and less wordy but that's not allowed, haha.

I think need correction

For a right triangle: a2+b2=c2a2+b2=c2.
For an acute (a triangle that has all angles less than 90°) triangle: a2+b2>c2
For an obtuse (a triangle that has an angle greater than 90°) triangle: a2+b2<c2
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Ans B
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AE wrote:
Ans B

Since, (angle b+ angle c )<90 degree. So, (b squared + C squared)<e squared. Similarly, ( angle a+angle d)>90 degree. So, ( a squared + d squared)>e squared. therefore, B is greater.
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Re: Which is greater b^2+2ad or (a+d)^2 - c^2 [#permalink]  27 Feb 2019, 08:34
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Try to look at this problem in a very simple way.

You have 2 triangles, one angle for each is given.

Triangle 1 with sides A and D and an angle 88 will have a greater angle at its disposition when compares to the other triangle with sides B and C and an angle of 92.

this inherently means, A+D > B+C

from the equations given, simplify and get it to this form and voila, you have an answer! ------seeeeeeee!!!!(C)
Re: Which is greater b^2+2ad or (a+d)^2 - c^2   [#permalink] 27 Feb 2019, 08:34
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