How many rectangles can be formed by taking the four vertices of 18-si

Question

How many rectangles can be formed by taking the four vertices of 18-sided regular polygon?

A. 16
B. 24
C. 32
D. 36
E. 3060

Are You Up For the Challenge: 700 Level Questions

exc4libur · Accepted Answer

straight lines: C(18,2)
triangles: C(18,3)
quadrilaterals: C(18,4)=18!/4!14!=3060
rectangle (4 right angles): 9!/2!=36

imagine the polygon in half and choose any two-points, from that half;
now, the number of ways we can choose the other two-points from opposite half is 1;
so, 18-sided polygon's half has 9 vertices to choose any two-points from: 9C2
and the number of ways to select the opposite vertices: 1
total: 9C2*1=36

Ans (D)

rajatchopra1994 · Answer

Thanks  Bunuel  ,

Explanation:

Let take an example for Hexagon, it will 3 rectangles (with 3 Diagonal pairs which are parallel)

9C2 = 36

IMO-D

Bunuel · Answer

This is not an easy question. Notice that we need rectangles, not simply quadrilaterals.

Subhrajyoti · Answer

18 sided polygon will have 18 vertices, using combination i.e. C( 18,4) = 3060 E

AabhishekGrover · Answer

Hi, can't this also be done by something like this:
(18*17)/4

18 because can select any of the 18 vertices and 17 because after selecting the vertex, one can choose any of the remaining 17 vertices and then multiply by 1 because for the remaining 2 vertices there is just one possibility.

Divide by 4 for repetition, please let me know where I'm going wrong

BALAGURU · Answer

For making a rectangle we need 4 points ( at 90 deg angle when connected)
so we have to divide the polygon with 9 points on one side ( say 1,2,3..9) and 9 on the other
Now, from this 9 points 
first select point 1, the combinations can be (1,2) (1,3) (1,4) ....(1,9) -----> 8 sets of rectangle 
Note that we have the corresponding other points in the other 9 points (10, 11, ... 18)
Total we have 8 rectangle

similarly, for point 2, the combinations can be (2,3) (2,4) ...(2,9) ------> 7 sets of rectangle 
same way we will have for all the points up to point 8, for which we have 1 rectangle

8+7+...1 =36

bumpbot · Answer

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.

How many rectangles can be formed by taking the four vertices of 18-si

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

How many rectangles can be formed by taking the four vertices of 18-si

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards