A regular octagon ABCDEFGH has an area of one square unit. What is the

Question

https://gmatclub.com/forum/download/file.php?id=47020 A regular octagon ABCDEFGH has an area of one square unit. What is the area of the rectangle ABEF? (A) 1 - \frac{\sqrt{2}}{2} (B) \frac{\sqrt{2}}{4} (C) \sqrt{2} - 1 (D) 1/2 (E) \frac{1 + \sqrt{2}}{4} 50ef8af0ec716cfeea89017a030c7d57d3b40e4c.png

itsanujsingh · Answer

Let each side of octagon be x

Area of octagon=2 (1+rt2)* x^2
As area given is 1 sq unit
We will get , x^2=1/(2(1+rt2)) {i}

Also to find area of rectangle ABFE, we need to find BF
For this extend AB and DC to meet at M,
Triangle CMB is a isosceles rt angle triangle, where CM=MB and BC=x
So CM =x/rt2
Similarly by extending CD and FE to meet at N we can find DN=x /rt2

Now BE=CM+CD+DN=x*(1+rt2)
And area ABEF= BE×AB=x^2(1+rt2)
Pitting value of x^2 from {i}

We get area ABEF=1/2

Posted from my mobile device

Archit3110 · Answer

Join AF & BE we see a rectangle been formed whose area would be equal to the 8 ∆ which are formed by joining opposite sides of the octagon 
so area of the rectangle = sum of 8 ∆ 
each side of length and breadth has 4 ∆ so ratio ; 4/8 = 1/2 
IMO D

replying to PM ;  NamGup  ; 
please see attachment..

Fdambro294 · Answer

Each Interior Angle of a Regular Octagon = 135 degrees

If you draw 2 Straight Parallel Lines from Points A to F and Points B to E you will have the Rectangle in the Middle of the Octagon with 90 degree Interior Angles.

If you draw 2 more Straight Parallel Lines from Points H to C and Points G to D you will have another Rectangle in the Middle of the Regular Octagon with 90 Degree Interior Angles

this will create 4 Right Triangles on the Corners of the Regular Octagon, with the following Hypotenuses -
AH
BC
DE
GF

The Area of the Regular Octagon will now be sub-divided into:

5 Squares with the Side of Each Square = Side of the Regular Octagon

+

4 Isosceles 45/45/90 Right Triangles with the Hypotenuse = Side of the Regular Octagon

Let the Equal Side of the Regular Octagon = A

Area of the Rectangle ABFE = 3 * (Area of our created Square with Side A) = 3 * (A * A) = 3 * (A)^2 = ?

Area of the Regular Octagon =

(5 Squares) * (A)^2

+

(4 Right Isosceles Triangles) *

=

1

__________________

5(A)^2 + 4 * (A^2 / 4) = 1

5(A)^2 + (A)^2 = 1

6(A)^2 = 1

the Question is asking for: What is 3(A)^2 = ?

if 6(A)^2 = 1, then HALF of this Area will be 1/2 = 3(A)^2

1/2
(D)

CrackverbalGMAT · Answer

A regular octagon ABCDEFGH has an area of one square unit. What is the area of the rectangle ABEF?

A regular octagon can be divided into 8 equal triangles as shown in figure.
Assume that each side of a regular octagon is a and the height of each triangle is h.

Area of regular octagon = 8 * area of one triangle = 8 * ½ * a * h= 4* a * h

We know that Area of regular octagon is given as 1 square unit
4* a * h =1 
A* h =1/4

Area of the rectangle ABEF = a* 2h = 2* ¼ = ½

Option D is the answer.

Thanks,
Clifin J Francis,
GMAT Quant SME

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

A regular octagon ABCDEFGH has an area of one square unit. What is the

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

A regular octagon ABCDEFGH has an area of one square unit. What is the

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