I assume the question is asking how many UNIQUE rectangles you can make. Bcause AB and BC are parallel to the y - and x - axes respectively, the rectangles will vary in size depending on where point C is.

That means the number of variations = the number of possibilities for point C. Number of x coordinates: 11 (0-10); number of y coordinates: 11. 11*11 = 121

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

Number of ways to select \(x_1\) = 11 (any value from 0 to 10)
Number of ways to select \(x_2\) = 10 (any value from 0 to 10, except the value we choose for \(x_1\))
Number of ways to select \(y_1\) = 11 (any value from 0 to 10)
Number of ways to select \(y_2\) = 10 (any value from 0 to 10, except the value we choose for \(y_1\))

Total possible rectangles = 11*10*11*10 = 12100

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.