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805+ (Hard)|   Geometry|      
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Bunuel
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In the diagram above, figure ABCD is a square with an area of 4.5 in^2 02 Jun 2015, 06:45
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In the diagram above, figure ABCD is a square with an area of 4.5 in^2. If the ratio of the length of DQ to the length of QB is 1 to 2, what is the length of QC, in inches?


A. \(\frac{\sqrt{10}}{2}\)

B. \(\frac{\sqrt{14}}{2}\)

C. \(2\sqrt{2}\)

D. \(2\sqrt{3}\)

E. \(2\sqrt{5}\)

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In the diagram above, figure ABCD is a square with an area of 4.5 in^2 04 Jun 2015, 10:07
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since the area is 4.5 or 45/10, each side of the square=\(\sqrt{\frac{45}{10}}\) = \(\frac{3}{\sqrt{2}}\)
With sides \(\frac{3}{\sqrt{2}}\), diagonal will be 3..
join C with centre of diag at T.. CT will be 1.5...and QT=DT-DQ=1.5-1=0.5..
now in triangle CTQ, CQ = \(\sqrt{(0.5)^2+(1.5)^2}\) = \(\sqrt{\frac{1}{4}+\frac{9}{4}}\)
=\(\sqrt{10}/2\)...
ans A
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In the diagram above, figure ABCD is a square with an area of 4.5 in^2 02 Jun 2015, 10:01
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Lets say DQ = x, QB = 2x
Diagonal = 3x
Side of the square = 3x/√2
Area = (3x/√2)^2 = 9x^2/2 = 4.5
9x^2 = 9; x =1
DQ = 1; QB = 2 ; Diagonal = 3
Side of square = 3/√2

See the attached figure.
Since BD is diagonal, angle BDC is 45 degree.
If you drop a perpendicular from Q to base CD you get a 45-45-90 triangle.
From here it is just application of Pythagoras theorem.
QC = sqrt {(1/√2)^2 + (2/√2)^2
= √10 / 2

Answer A
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In the diagram above, figure ABCD is a square with an area of 4.5 in^2 04 Jun 2015, 09:21
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I got the first half of when you mentioned although post this, I used normal Pythogoras

QD^2 +QC^2 = CD^2
1 +QC^2=9/2
QC => \sqrt{7/2}

what am I doing wrong here??


askhere
Lets say DQ = x, QB = 2x
Diagonal = 3x
Side of the square = 3x/√2
Area = (3x/√2)^2 = 9x^2/2 = 4.5
9x^2 = 9; x =1
DQ = 1; QB = 2 ; Diagonal = 3
Side of square = 3/√2

See the attached figure.
Since BD is diagonal, angle BDC is 45 degree.
If you drop a perpendicular from Q to base CD you get a 45-45-90 triangle.
From here it is just application of Pythagoras theorem.
QC = sqrt {(1/√2)^2 + (2/√2)^2
= √10 / 2

Answer A
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In the diagram above, figure ABCD is a square with an area of 4.5 in^2 04 Jun 2015, 10:30
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I seem to get lost here a bit.
What I did first was to get the diagonal DB, since we know the area of 4.5 inch.

area of square = side^2
so the side would be sqrt (4.5)
So Diagonal is sqrt( 4.5 )^2 + sqrt( 4.5 )^2 = sqrt ( 9 ) = 3

Since DQ ratio is 1:2, DQ has to be 1
This is the point I got lost on.

After some reading on another answer I thought the perpendicular line would help, but then i got stuck on how to simplify the square roots in a fraction.
If someone could also help me out on some reading material regarding square roots and fractions and the reciprocals of them that would be great.

Because why is (From Askhere)
QC = sqrt {(1/√2)^2 + (2/√2)^2
= √10 / 2
I mean when raising these to the power you would square root of 3 right?
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In the diagram above, figure ABCD is a square with an area of 4.5 in^2 04 Jun 2015, 11:11
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QC is not perpendicular to DB.(see ratio DQ/QB is 1/2)

so you can not apply Pythagoras here.

Regards

jimmy

kelvind13
I got the first half of when you mentioned although post this, I used normal Pythogoras

QD^2 +QC^2 = CD^2
1 +QC^2=9/2
QC => \sqrt{7/2}

what am I doing wrong here??


askhere
Lets say DQ = x, QB = 2x
Diagonal = 3x
Side of the square = 3x/√2
Area = (3x/√2)^2 = 9x^2/2 = 4.5
9x^2 = 9; x =1
DQ = 1; QB = 2 ; Diagonal = 3
Side of square = 3/√2

See the attached figure.
Since BD is diagonal, angle BDC is 45 degree.
If you drop a perpendicular from Q to base CD you get a 45-45-90 triangle.
From here it is just application of Pythagoras theorem.
QC = sqrt {(1/√2)^2 + (2/√2)^2
= √10 / 2

Answer A
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In the diagram above, figure ABCD is a square with an area of 4.5 in^2 04 Jun 2015, 20:39
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[quote="kelvind13"]I got the first half of when you mentioned although post this, I used normal Pythogoras

QD^2 +QC^2 = CD^2
1 +QC^2=9/2
QC => \sqrt{7/2}

what am I doing wrong here??


Hi kelvind13,

To apply Pythagoras theorem it has to be a right triangle.
You have assumed Traingle DQC to be a right triangle which is wrong. This is where you went wrong.
Hope you understood.
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In the diagram above, figure ABCD is a square with an area of 4.5 in^2 08 Jun 2015, 05:25
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Bunuel

In the diagram above, figure ABCD is a square with an area of 4.5 in^2. If the ratio of the length of DQ to the length of QB is 1 to 2, what is the length of QC, in inches?

A. \(\frac{\sqrt{10}}{2}\)
B. \(\frac{\sqrt{14}}{2}\)
C. \(2\sqrt{2}\)
D. \(2\sqrt{3}\)
E. \(2\sqrt{5}\)

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The attachment 2015-06-02_1741.png is no longer available
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MANHATTAN GMAT OFFICIAL SOLUTION:

If the area of the square is 4.5 and the side length is s, then \(s^2=\frac{9}{2}\) and \(s =\frac{3\sqrt{2}}{2}\)

Triangle BCD is a 45–45–90 right triangle, so the ratio of its sides is \(s : s : s \sqrt{2}\). Thus, DB (the hypotenuse) is \(s \sqrt{2}=\frac{3\sqrt{2}}{2}*\sqrt{2}=3\). Furthermore, if DB is 3, and the ratio DQ : QB is 1 : 2, it must be true that DQ = 1 and QB = 2.

To be able to use Pythagorean Theorem, we must create a right triangle with QC as a side by drawing a new line. Such a right triangle will be similar to triangle BCD. Here are two solutions, the only difference being how one draws the extra line:



The correct answer is A.
Show Spoiler
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2015-06-08_1619.png
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2015-06-08_1619_001.png
2015-06-08_1619_001.png [ 24.45 KiB | Viewed 11428 times ]
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In the diagram above, figure ABCD is a square with an area of 4.5 in^2 02 Jul 2015, 04:17
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Hey, can someone pls explain what is wrong with the below approach?

QD^2 +QC^2 = CD^2
1 +QC^2=9/2
QC => \sqrt{7/2}
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In the diagram above, figure ABCD is a square with an area of 4.5 in^2 14 Mar 2017, 18:13
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chetan2u
since the area is 4.5 or 45/10, each side of the square=\(\sqrt{\frac{45}{10}}\) = \(\frac{3}{\sqrt{2}}\)
With sides \(\frac{3}{\sqrt{2}}\), diagonal will be 3..
join C with centre of diag at T.. CT will be 1.5...and QT=DT-DQ=1.5-1=0.5..
now in triangle CTQ, CQ = \(\sqrt{(0.5)^2+(1.5)^2}\) = \(\sqrt{\frac{1}{4}+\frac{9}{4}}\)
=\(\sqrt{10}/2\)...
ans A
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Hello Chetan,

Can you please explain how did you concluded BT as 1.5?
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In the diagram above, figure ABCD is a square with an area of 4.5 in^2 27 Dec 2017, 13:07
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Dear Bunuel
One thing that I didn't understood from the second Manhattam Official Solution is:
why DQ = 2, as the ratio of both triangles is 1:3?
Could you clarify that to me?
Thanks!

Bunuel
Bunuel

In the diagram above, figure ABCD is a square with an area of 4.5 in^2. If the ratio of the length of DQ to the length of QB is 1 to 2, what is the length of QC, in inches?

A. \(\frac{\sqrt{10}}{2}\)
B. \(\frac{\sqrt{14}}{2}\)
C. \(2\sqrt{2}\)
D. \(2\sqrt{3}\)
E. \(2\sqrt{5}\)

Show Spoiler
Attachment:
2015-06-02_1741.png

MANHATTAN GMAT OFFICIAL SOLUTION:

If the area of the square is 4.5 and the side length is s, then \(s^2=\frac{9}{2}\) and \(s =\frac{3\sqrt{2}}{2}\)

Triangle BCD is a 45–45–90 right triangle, so the ratio of its sides is \(s : s : s \sqrt{2}\). Thus, DB (the hypotenuse) is \(s \sqrt{2}=\frac{3\sqrt{2}}{2}*\sqrt{2}=3\). Furthermore, if DB is 3, and the ratio DQ : QB is 1 : 2, it must be true that DQ = 1 and QB = 2.

To be able to use Pythagorean Theorem, we must create a right triangle with QC as a side by drawing a new line. Such a right triangle will be similar to triangle BCD. Here are two solutions, the only difference being how one draws the extra line:



The correct answer is A.
Show Spoiler
Attachment:
2015-06-08_1619.png
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2015-06-08_1619_001.png
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In the diagram above, figure ABCD is a square with an area of 4.5 in^2 27 Dec 2017, 23:24
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robertops
Dear Bunuel
One thing that I didn't understood from the second Manhattam Official Solution is:
why DQ = 2, as the ratio of both triangles is 1:3?
Could you clarify that to me?
Thanks!

Bunuel
Bunuel

In the diagram above, figure ABCD is a square with an area of 4.5 in^2. If the ratio of the length of DQ to the length of QB is 1 to 2, what is the length of QC, in inches?

A. \(\frac{\sqrt{10}}{2}\)
B. \(\frac{\sqrt{14}}{2}\)
C. \(2\sqrt{2}\)
D. \(2\sqrt{3}\)
E. \(2\sqrt{5}\)

Show Spoiler
Attachment:
2015-06-02_1741.png

MANHATTAN GMAT OFFICIAL SOLUTION:

If the area of the square is 4.5 and the side length is s, then \(s^2=\frac{9}{2}\) and \(s =\frac{3\sqrt{2}}{2}\)

Triangle BCD is a 45–45–90 right triangle, so the ratio of its sides is \(s : s : s \sqrt{2}\). Thus, DB (the hypotenuse) is \(s \sqrt{2}=\frac{3\sqrt{2}}{2}*\sqrt{2}=3\). Furthermore, if DB is 3, and the ratio DQ : QB is 1 : 2, it must be true that DQ = 1 and QB = 2.

To be able to use Pythagorean Theorem, we must create a right triangle with QC as a side by drawing a new line. Such a right triangle will be similar to triangle BCD. Here are two solutions, the only difference being how one draws the extra line:



The correct answer is A.
Show Spoiler
Attachment:
2015-06-08_1619.png
Attachment:
2015-06-08_1619_001.png
Attachment:
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It's a typo there. DQ should be 1.
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In the diagram above, figure ABCD is a square with an area of 4.5 in^2 28 Dec 2017, 01:24
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Bunuel

In the diagram above, figure ABCD is a square with an area of 4.5 in^2. If the ratio of the length of DQ to the length of QB is 1 to 2, what is the length of QC, in inches?

A. \(\frac{\sqrt{10}}{2}\)
B. \(\frac{\sqrt{14}}{2}\)
C. \(2\sqrt{2}\)
D. \(2\sqrt{3}\)
E. \(2\sqrt{5}\)

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2015-06-02_1741.png
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Hi Bunuel, chetan2u,

Can you please help me to clarify my doubt.

From the given info on the question DB = 3, DQ =1 & QB = 2. Let P be mid point of QB so, DQ = QP = PB = 1. As each side are equal, DQ, QP & PB will subtend an equal angle at C so Angle QCP = Angle QCD = Angle PCB = 30. Now, as per property of Triangle for Triangle QCD \(\frac{QD}{Sin30} = \frac{QC}{Sin45} = \frac{CD}{Sin75}\). By using this logic I am getting a different answer for QC.

Can you please guide as to where I am going wrong.
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In the diagram above, figure ABCD is a square with an area of 4.5 in^2 28 Dec 2017, 02:17
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Bunuel

In the diagram above, figure ABCD is a square with an area of 4.5 in^2. If the ratio of the length of DQ to the length of QB is 1 to 2, what is the length of QC, in inches?

A. \(\frac{\sqrt{10}}{2}\)
B. \(\frac{\sqrt{14}}{2}\)
C. \(2\sqrt{2}\)
D. \(2\sqrt{3}\)
E. \(2\sqrt{5}\)

Show Spoiler
Attachment:
2015-06-02_1741.png

Hi Bunuel, chetan2u,

Can you please help me to clarify my doubt.

From the given info on the question DB = 3, DQ =1 & QB = 2. Let P be mid point of QB so, DQ = QP = PB = 1. As each side are equal, DQ, QP & PB will subtend an equal angle at C so Angle QCP = Angle QCD = Angle PCB = 30. Now, as per property of Triangle for Triangle QCD \(\frac{QD}{Sin30} = \frac{QC}{Sin45} = \frac{CD}{Sin75}\). By using this logic I am getting a different answer for QC.

Can you please guide as to where I am going wrong.
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You have gone wrong in taking all three angles as 30..
If you have got angle QCD as 75 so QCB as 180-75 =105...
Now if I take the centre triangle QCP, it should be isosceles as the side triangles QCD and PCB are similar..
So the two angles become 105*2=210>180.. not possible..
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In the diagram above, figure ABCD is a square with an area of 4.5 in^2 17 Jul 2022, 06:35
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