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In the trapezoid above, with sides a, b, c, and 1, what is the value o

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Math Expert
Joined: 02 Sep 2009
Posts: 46146
In the trapezoid above, with sides a, b, c, and 1, what is the value o [#permalink]

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30 Apr 2015, 04:17
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95% (hard)

Question Stats:

34% (01:37) correct 66% (02:03) wrong based on 256 sessions

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In the trapezoid above, with sides a, b, c, and 1, what is the value of ac ?

(1) a + c = b

(2) a – c = 1

Kudos for a correct solution.

Attachment:

lnzFUe1NsWpRHYetrW_BAMHT3M0ml9ssJU6sexynIY0SxO2MrXLRX71d4EjIqD-ITXHoVyRCXvhKECDrtvrdwtt0Q9b1h5-W_LSClrh9liw7pUognufJTS-OvSXIC20UfQ.png [ 2.03 KiB | Viewed 3025 times ]

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Manager
Joined: 01 Jan 2015
Posts: 56
Re: In the trapezoid above, with sides a, b, c, and 1, what is the value o [#permalink]

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01 May 2015, 13:46
1
ST 1 A+C=B
Draw a line parallel to side-measure 1 from C TO A
Now for the right angle triangle 1^2 + (a-c)^2 = b^2
on solving this we'll get the value of ac ST 1 SUFF
ST 2 A-C=1
Senior Manager
Joined: 28 Feb 2014
Posts: 295
Location: United States
Concentration: Strategy, General Management
In the trapezoid above, with sides a, b, c, and 1, what is the value o [#permalink]

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01 May 2015, 16:17
1
what is the value of ac ?
We have to find a way to create an equation from the trapezoid that will draw an appropriate relationship between the statements and the information given.
Drawing a line parallel to the side that has a value of 1, we create a right triangle with a-c and 1 as its sides and b as its hypotenuse.
Using pythagorean theorem, we obtain the equation (a-c)^2 + 1^2 = b^2
Decomposed, we obtain the equation a^2 - 2ac + c^2 + 1 = b^2

(1) a + c = b
If we square both sides of this entity, it become a^2 + 2ac + c^2 = b^2
we can then use that equation and subtract it from a^2 - 2ac + c^2 + 1 = b^2
ac in this case equals 1/4
Sufficient

(2) a – c = 1
Even after squaring this we are unable to isolate ac into a specific value
Insufficient

Intern
Joined: 06 Apr 2015
Posts: 35
Re: In the trapezoid above, with sides a, b, c, and 1, what is the value o [#permalink]

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04 May 2015, 01:42

Ted21 wrote:
ST 1 A+C=B
Draw a line parallel to side-measure 1 from C TO A
Now for the right angle triangle 1^2 + (a-c)^2 = b^2
on solving this we'll get the value of ac ST 1 SUFF
ST 2 A-C=1
Senior Manager
Joined: 28 Feb 2014
Posts: 295
Location: United States
Concentration: Strategy, General Management
Re: In the trapezoid above, with sides a, b, c, and 1, what is the value o [#permalink]

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04 May 2015, 17:38
1
kelvind13 wrote:

Ted21 wrote:
ST 1 A+C=B
Draw a line parallel to side-measure 1 from C TO A
Now for the right angle triangle 1^2 + (a-c)^2 = b^2
on solving this we'll get the value of ac ST 1 SUFF
ST 2 A-C=1

Statement 1:
a+c=b
square both sides of the above equation. This then becomes a^2 + 2ac + c^2 = b^2

a^2 + 2ac + c^2 = b^2
-(a^2 - 2ac + c^2 + 1 = b^2)

When we subtract the two equations it becomes

a^2 + 2ac + c^2 = b^2
-a^2 + 2ac - c^2 - 1 = -b^2

which equals
4ac-1=0
4ac=1
ac=(1/4)

Hopefully this helps
Intern
Joined: 13 Sep 2014
Posts: 27
Concentration: Entrepreneurship, Strategy
Re: In the trapezoid above, with sides a, b, c, and 1, what is the value o [#permalink]

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02 Aug 2016, 11:38
Bunuel,

Can you please post the complete solution?

Thanks
Akash
Director
Joined: 05 Mar 2015
Posts: 969
In the trapezoid above, with sides a, b, c, and 1, what is the value o [#permalink]

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02 Aug 2016, 18:56
1
AkashKashyap wrote:
Bunuel,

Can you please post the complete solution?

Thanks
Akash

Akash

(1) a+c=b
sq. both sides
a^2+c^2+2ac=b^2-----------(A)
also from fig the cutout upper right angled triangle having hypotenuse b we can write
(a-c)^2+1^2=b^2
a^2+c^2-2ac+1=b^2-----------(B)
now equate (A) & (B)

a^2+c^2+2ac=a^2+c^2-2ac+1
ac=1/4
suff...

(2) a-c=1
we can only calculate b=sq rt 2 nothing else
insuff...

Ans A
Attachments

lnzFUe1NsWpRHYetrW_BAMHT3M0ml9ssJU6sexynIY0SxO2MrXLRX71d4EjIqD-ITXHoVyRCXvhKECDrtvrdwtt0Q9b1h5-W_LSClrh9liw7pUognufJTS-OvSXIC20UfQ.png [ 4.96 KiB | Viewed 1552 times ]

Manager
Joined: 28 Jul 2016
Posts: 126
Re: In the trapezoid above, with sides a, b, c, and 1, what is the value o [#permalink]

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01 Apr 2017, 09:28
Bunuel wrote:

In the trapezoid above, with sides a, b, c, and 1, what is the value of ac ?

(1) a + c = b

(2) a – c = 1

Kudos for a correct solution.

Attachment:
lnzFUe1NsWpRHYetrW_BAMHT3M0ml9ssJU6sexynIY0SxO2MrXLRX71d4EjIqD-ITXHoVyRCXvhKECDrtvrdwtt0Q9b1h5-W_LSClrh9liw7pUognufJTS-OvSXIC20UfQ.png

Great question! thanks for sharing this.
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 5575
GMAT 1: 800 Q59 V59
GPA: 3.82
Re: In the trapezoid above, with sides a, b, c, and 1, what is the value o [#permalink]

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01 Apr 2017, 10:10
1
Bunuel wrote:

In the trapezoid above, with sides a, b, c, and 1, what is the value of ac ?

(1) a + c = b

(2) a – c = 1

Kudos for a correct solution.

Attachment:
lnzFUe1NsWpRHYetrW_BAMHT3M0ml9ssJU6sexynIY0SxO2MrXLRX71d4EjIqD-ITXHoVyRCXvhKECDrtvrdwtt0Q9b1h5-W_LSClrh9liw7pUognufJTS-OvSXIC20UfQ.png

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.

There are 3 variables and 1 hidden equation from Pythagoras theorem, which is $$(a-c)^2 + 1 = b^2$$. Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. However, this question is not about a single variable, but about ab. We need to look into any relation about $$ab$$.

Condition 1)

$$(a+c)^2 = b^2 = (a-c)^2 + 1$$
$$(a+c)^2 = (a-c)^2 + 1$$
$$a^2 + 2ac + c^2 = a^2 -2ac + c^2 + 1$$
$$2ac = -2ac + 1$$
$$4ac = 1$$
$$ac = 1/4$$

This is sufficient.

Condition 2)

We have $$(a-c)^2 + 1 = b^2$$.
$$b^2 = 2$$
We can not declare the value of $$ac$$.

Thus, this is not sufficient.

Therefore, the correct answer is A.
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Re: In the trapezoid above, with sides a, b, c, and 1, what is the value o   [#permalink] 01 Apr 2017, 10:10
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