A rectangle, ABCD, is to be constructed in an XY-plane such that AB is

Hey Nick I have question,

Every square is technically a rectangle, should we be considering that in such questions?

rheam25

Yes you must consider square too among rectangles.

Yup you have to consider all possible rectangles(including squares). Never ignore the squares unless it's given (that would be an interesting question tho).

GMATinsight Damn! you're fast.

[quote="nick1816"]

Coordinates of A = \((x_1,y_1)\)
Coordinates of B = \((x_2,y_1)\)
Coordinates of C = \((x_2,y_2)\)
Coordinates of D = \((x_1,y_2)\)


Hi Nick,

Even though I understand your answer method, should't the y coordinate of B become y_2 and not y_1, give AB is parallel to y axis.

Hello nick1816
Why is it not 11C2*11C2?
Thank you in advance.

nick1816
Bunuel

Since these points must form a rectangle, CD must be parallel to AB, and AD must be parallel to BC. Therefore, D would be restricted by the values of A and C, I would think that you would not have any options for the value of D. Can you please explain?
Thank you!

I second that. please explain this..

41396302717
Because order matters here, as the name of vertices are given.

juliahamm24
You're absolutely right. If we fix A and C, B and D would also get fixed, since sides are parallel to x-axis or y-axis. But to fix A and C, we have to fix \(x_1\),\(x_2\),\(y_1\) and \(y_2\). All of them(x_1,x_2,y_1 and y_2) can take values independent of each other.

If you still have any doubt, you can ask

Step 1) Bound the Region in which the Rectangles can be Formed

Horizontal Line Y = 10
and
Vertical Line X = 10

Bound the Region in QUADRANT 1 in which we can draw these Rectangles (with the Integer Coordinates on the 2 Lines included)


Step 2) Realize the Picture of Each Rectangle and the Placement of Each Vertex


In order for AB to Parallel to the Y-Axis and BC to Parallel to the X-Axis ----- Vertex B of the Rectangle must be in the 1 of the LEFT VERTICES of Each Rectangle Drawn


Vertex A must lie on the SAME Vertical Line as Vertex B



Because Side BC is Parallel to the X-Axis: Vertex C must lie on the SAME Horizontal Line as Vertex B



Finally, to complete any given rectangle, Vertex D must lie on the SAME Vertical Line as Vertex C and the SAME Horizontal Line as Vertex A




(3rd) Using the Constraints, COUNT How Many Rectangles we can draw within the Bounded Region


-1- Start with Selecting the Coordinates for Vertex B (Xb ; Yb)

The Possible Integer Values to select as Coordinates are from [0 - to - 10], inclusive = 11 Values

The X Coordinate of B and Y Coordinate of B can each take ANY Arrangement of these 11 Integer Values

Xb -- 11 Options
Yb --- 11 Options




-2- Vertex C (Xc ; Yc)


Yc - Coordinate:
Because Vertex C must lie on the SAME, Horizontal Line as Vertex B (Rule: All Rectangles have OPP. Parallel Sides with each Adjacent Angle = 90 degrees) ----

for Any Particular Arrangement of Vertex B’s Coordinates, Vertex C will have to take the SAME Y Coordinate as Point B.

Yc --- 1 “Fixed” Option


Xc Coordinate:
Vertex C can move along the X-Axis and take any Integer Value that is NOT Already taken by Vertex B

Xc --- 11 Possibilities - 1 Chosen for Vertex B = 10 Options





Vertex A:
A =(Xa ; Ya)


Xa - Coordinate:
Since Vertex A must lie on the SAME Vertical Line as Vertex B, Vertex A's X-COORDINATE must take the SAME X-Coordinate that was chosen for Vertex B for any given Arrangement. Similar to a "Palindrome Integer Q-Type", the Option becomes FIXED.

Xa ----- 1 “Fixed” Option


Ya - Coordinate:
The Y-Coordinate of Vertex A can take ANY Integer Value that has NOT been Chosen for Vertex B Already

Ya ----- 11 Possible Integers - 1 Chosen for Vertex B = 10 Options



Vertex D
D = (Xd ; Yd)

For any given Arrangement of the 3 Vertices we choose above, Vertex D must always be FIXED.

In order to complete any given Rectangle with Opposite Sides Parallel and Adjacent Angles of
90 deg. each:

The X-Coordinate will have to take the Same X-Coordinate as Vertex C.

The Y-Coordinate will have to take the Same Y-Coordinate as Vertex A

Xd ----- 1 “Fixed” Option

Yd ---- 1 “Fixed” Option



Finally:

The Total Number of Possible Arrangements of Coordinates that form Rectangles =

(11 * 11) * (10 * 1) * (1 * 10) * (1 * 1) = 121 * 100 = 12, 100


-E-
12, 100 Possible Rectangles can be drawn

Thank You, will keep this in mind while trying other questions.

Hello,nick1816
what do you mean by the order matters here
Why can't we do 11c2*11c2?
we have 11 points for x of which we need two and 11 points for y of which we need 2
can you please explain

First point to be chosen from 11*11=121 total points.

Next, let's say the northeast point to be chosen from 10 different points, and one of the lower point to be chosen from 10 different points. The 4th point is chosen automatically, for the condition that this is a rectangle to hold.

According to this: 121*10*12=12,100.

But what if the problem asked how many unique rectangles can we form? should we divide by 4 in that case? Since we have 4 different points to choose from?

Bunuel, chetan2u, Your explanations would be very helpful.

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