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Geometry Questions.....

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Geometry Questions..... [#permalink] New post 24 Oct 2006, 23:32
Hi All, a couple of questions:Please answer them in detail. I am really confused here....

Q1:
Is the area of the rectangle in Fig. A equal to 50?

(1) z2 = 100

(2) x = y + 2


Q2:
In the fig B, the length of segment xz is 24. What is the circumference of the circle with diameter wy?

(1) The length of segment wx is 40.

(2) The length of segment xy is 30.
Attachments

Fig B-Q6 Math 5-6 Adv Drill.JPG
Fig B-Q6 Math 5-6 Adv Drill.JPG [ 6.58 KiB | Viewed 650 times ]

Fig A Q3-Math-5-6 Adv Drill.JPG
Fig A Q3-Math-5-6 Adv Drill.JPG [ 3.61 KiB | Viewed 650 times ]

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Re: Geometry Questions..... [#permalink] New post 25 Oct 2006, 00:08
usman7 wrote:
Hi All, a couple of questions:Please answer them in detail. I am really confused here....

Q1:
Is the area of the rectangle in Fig. A equal to 50?

(1) z2 = 100

(2) x = y + 2


Q2:
In the fig B, the length of segment xz is 24. What is the circumference of the circle with diameter wy?

(1) The length of segment wx is 40.

(2) The length of segment xy is 30.


Q1 - 1 is insufficient because knowing the length of the diagonal of a rectangle is insufficient to calculate the area.

2 - alone is again insufficient. Area of a rectangle is the product of the lengths of its sides. Here only a relation between the sides is expressed not the actual value.

Now look at 1 and 2 in combination - since we know diagonal and we know the relation of the lengths of the two sides we can set up an eqn like this

(y+2)^2 + y^2 = z^2 = 100. so (y+2)^2 + y^2 = 100. Solving this for Y and then using x = y + 2 we will get X. We get the area.

So we need both choices to arrive at an answer hence C.

For Q2 -

We again need both choices to determine the circumference.
C = 2*pi*r where r = radius or C = pi*D where D = diameter.


D = yz + zw.

We are given - xz = 24. Using xz and choice 1 alone we can determine zw.

Using xz and choice 2 alone we can determine yz.

Neither stmt alone is sufficient. Combining we get the answer. So ans is C.
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 [#permalink] New post 25 Oct 2006, 00:13
For the 1Q i would choose

A

1) z^2=x^2+y^2....
so we have

x^2+y^2=100

there is no way the area of that rectangular is 50.....

For the 2Q i would choose

C

Both statements coupled with the info given in the question stem provide data for two of the side of the triangle.... therefore in either case we can calculate the length of the remaining side.... which will provide us with the length of the diameter....
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 [#permalink] New post 25 Oct 2006, 05:05
My Answers will be A for question 1

x^2+y^2 =100 implies that area of the rectangle will be 48 (6,8,10)

For question 2

My pick is D.

As Either statement gives us value of one side of triangle the value of other 2 required sides can be found using pythagoras and similarity of triangles.

What are the official answers.
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 [#permalink] New post 25 Oct 2006, 06:11
The key to the second question is to know that angle wxy is 90 degree.
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 [#permalink] New post 25 Oct 2006, 08:22
Thanks Guys....but i guess we share some common misconcepts :)....

The Ans to Q1 is C ...but guys the same question looms i.e. if the diagonal is 10 and its a rectangle, can we assume that its a 3-4-5 triangle.......so please if any one can clear this? do we need to have atleast two sides to determine if its 3-4-5 or any special triangle? I just feel, ive solved some questions by knowing only one side....but cant seem to trace those questions now...or maybe i am confusing it altogether.

The Ans to Q2 is D but i dont know how do we solve it with similarity...yogeshsheth, can you please explain it in detail. How do we know if a two triangles are similar? if you know of any link to such an article on similarity, it would be great if you can provide the link. i wasn't able to find any good stuff on it on the net.

Waiting for more explanation. Honghu your input would be highly welcomed.
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Explanation of Triangle Q [#permalink] New post 25 Oct 2006, 08:50
(1) Basic rule is similar trinagles side have same ratio of sides

Trinagle XZW is similar to trinagle ZXY we know side WX and WZ using pythogarus theorum we know ise XZ we know all sides of 1 trinaagle and one side of other trinagle. If u know ratiof of 1 side of a similar triangle and all three isdes of other trinagle u can fighure out rest of the two unkown sides

SO u can calculate ZY thus we know diameter

(2)USe similar trinagle property on this 1 too trinagle WXY and XZY are similar. We know 2 sides of 1 trinagle and 1 ide of 2nd trinagle

So we can derive all sides of both trinagle
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 [#permalink] New post 25 Oct 2006, 09:17
Q1.
We are required to find x and y to get area.
C is my answer.

Combining

x^2 + y^2 = z^2

x^2 + (x - 2)^2 = 100 (since x + y = 2 => y = x - 2)

2x^2 - 4x + 4 - 100 = 0 ( since (a-b)^2 = a^2 -2ab + b^2)

x^2 - 2x + 2 - 50 = 0
x^2 - 2x -48 = 0
(x - 8)(x + 6) = 0

x = 8 (since x cant be negative)
y = 6

C.
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 [#permalink] New post 25 Oct 2006, 09:22
Quote:
(1) Basic rule is similar trinagles side have same ratio of sides

Trinagle XZW is similar to trinagle ZXY we know side WX and WZ using pythogarus theorum we know ise XZ we know all sides of 1 trinaagle and one side of other trinagle. If u know ratiof of 1 side of a similar triangle and all three isdes of other trinagle u can fighure out rest of the two unkown sides

SO u can calculate ZY thus we know diameter

(2)USe similar trinagle property on this 1 too trinagle WXY and XZY are similar. We know 2 sides of 1 trinagle and 1 ide of 2nd trinagle

So we can derive all sides of both trinagle


So Damage, what I am not getting is...how do we know in the first place that WXZ and XYZ are similar triangles? Moreover, dimensions WXZ come out to be 24-32-40(3-4-5) and the multiplying factor is 8...now i know i sound dumb but please just explain it that how is length of the side YZ can be determined from it? What is the common ratio?
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Edited below [#permalink] New post 25 Oct 2006, 10:56
Removed look at following post

Last edited by Damager on 25 Oct 2006, 12:25, edited 1 time in total.
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 [#permalink] New post 25 Oct 2006, 10:58
Somehow it strikes me that the answer to the 1Q is not A......
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More details [#permalink] New post 25 Oct 2006, 12:24
XZW is similar to trinagle ZXY

(1) angles WZX = angle YZX = 90 degree

(2) WXY is 90 degrees because a diameter always makes 90 degree angle on cirumference

(3) angle WXY = 90 - angle ZXY
angle ZYX = 90 degree - ZXY

So angle WXY=angle ZYX
we already know angles WZX = angle YZX = 90 degree

Two of the three angles of respective trinagles are equal to each other. Since sum of angles in trinagle = 180, if two angles are eqal third will be equal too

190 - sum of other two angles of ZXY = 180 - sum of other two angles of a triangle XZW

Based on similarity

in rt trinagle XZW 40^2 = 24^2 + wz^2==> WZ=10

Since angle wxy = angle xyz sides WZ/XZ is ratio of sides = 40/24
similar traingle WZ/XZ=XZ/ZY==> ZY= 24*24/40= whatever number u know the diameter

Use same logic for 2nd case
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Re: More details [#permalink] New post 25 Oct 2006, 20:37
Damager wrote:
XZW is similar to trinagle ZXY

(1) angles WZX = angle YZX = 90 degree

(2) WXY is 90 degrees because a diameter always makes 90 degree angle on cirumference

(3) angle WXY = 90 - angle ZXY
angle ZYX = 90 degree - ZXY

So angle WXY=angle ZYX
we already know angles WZX = angle YZX = 90 degree

Two of the three angles of respective trinagles are equal to each other. Since sum of angles in trinagle = 180, if two angles are eqal third will be equal too

190 - sum of other two angles of ZXY = 180 - sum of other two angles of a triangle XZW

Based on similarity

in rt trinagle XZW 40^2 = 24^2 + wz^2==> WZ=10

Since angle wxy = angle xyz sides WZ/XZ is ratio of sides = 40/24
similar traingle WZ/XZ=XZ/ZY==> ZY= 24*24/40= whatever number u know the diameter

Use same logic for 2nd case


Very well explanied damager :-D
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 [#permalink] New post 25 Oct 2006, 22:31
Q1) for me (C) as well

Stat 1
z^2 = 100
<=> x^2 + y^2 = 100 (1)

Can we have x*y = 50 ? Let try to set :

x*y = 50
<=> 2*x*y = 100 (2)

(1) + (2)
<=> x^2 + 2*x*y + y^2 = 200
<=> (x+y)^2 = 200
=> x + y = sqrt(200) since x and y must be pisotive (they are distances)

It is also possible to fix x and y such that they respect the equation (1) but not (2) (x = 1 and y = sqrt(99))

INSUFF


Stat 2
x = y + 2

x and y can be any numbers that verify this equation but not necesserary x*y=50

INSUFF


Stat 1 and 2 combined

(y + 2)^2 + y^2 = 100
<=> 2*y^2 + 4*y + 4 = 100
<=> y^2 + 2*y + 2 = 50
<=> y^2 + 2*y - 48 = 0

Reducted Delta : 1+48 = 49 = (7)^2

y = -1 + 7 = 6
or
y = -1 - 7 = 8

Since a distance is always positive, y = 6 and so x = 8. Thus, x*y = 6*8 = 48.

SUFF
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Re: Geometry Questions..... [#permalink] New post 26 Oct 2006, 04:22
usman7 wrote:
Q1:
Is the area of the rectangle in Fig. A equal to 50?

(1) z2 = 100

(2) x = y + 2

C

x^2 + y^2 = z^2. -------> (1)

Statement 1
No information about x or y. INSUFF

Statement 2:
Substitute y+2 for x in (1) INSUFF

Together,

2y^2 + 4y + 4 = 100
Simplify to get
y = 6 or 8. Area = 48

Quote:
Q2:
In the fig B, the length of segment xz is 24. What is the circumference of the circle with diameter wy?

(1) The length of segment wx is 40.

(2) The length of segment xy is 30.

C
Not very sure about this, but here goes:

Statement 1 INSUFF
Statement 2 INSUFF

Together, we know that angle subscribed within a circle from the diameter is 90.
wx and xy are two legs of a right-angled triangle. wy is the hypotenuse of this triangle and should be 50. SUFF
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Re: More details [#permalink] New post 26 Oct 2006, 04:27
Damager wrote:
XZW is similar to trinagle ZXY

(1) angles WZX = angle YZX = 90 degree

(2) WXY is 90 degrees because a diameter always makes 90 degree angle on cirumference

(3) angle WXY = 90 - angle ZXY
angle ZYX = 90 degree - ZXY

So angle WXY=angle ZYX
we already know angles WZX = angle YZX = 90 degree

Two of the three angles of respective trinagles are equal to each other. Since sum of angles in trinagle = 180, if two angles are eqal third will be equal too

190 - sum of other two angles of ZXY = 180 - sum of other two angles of a triangle XZW

Based on similarity

in rt trinagle XZW 40^2 = 24^2 + wz^2==> WZ=10

Since angle wxy = angle xyz sides WZ/XZ is ratio of sides = 40/24
similar traingle WZ/XZ=XZ/ZY==> ZY= 24*24/40= whatever number u know the diameter

Use same logic for 2nd case

Great explanation, Damager.
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 [#permalink] New post 26 Oct 2006, 06:15
Thanks Damager...that was excellent.......now its pretty clear
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 [#permalink] New post 26 Oct 2006, 15:05
Great posts! Good stuff. :panel
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 [#permalink] New post 26 Oct 2006, 17:43
Q 2 - D
Great explanation by damager

One more thing we need to note. Here both the smaller triangle XZW and ZXY are similar to big triangle WXY for the same reason – all angles are same.
For bigger triangle, the right angle is WXY

From 1
So we can write, Hypotenuse / Base of smaller triangle XZW = Hypotenuse / Base of bigger triangle WXY
=> XW / WZ = WY / XW
=> 40 / squrt (40^2 – 24^2) = diameter / 40
=> Diameter = 40^2 / 32 = 50

From 2
Diameter can be calculated using same logic.
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 [#permalink] New post 27 Oct 2006, 10:35
Excellent anindyat...
  [#permalink] 27 Oct 2006, 10:35
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