CAT Practice : Geometry-Triangles

You are here: Home  CAT Questionbank   CAT Quant  Geometry: Triangles  Question 11
Octagons are one of the most fascinating forms of a polynomial.

Diagonals of octagon

Q.11: What is the ratio of longest diagonal to the shortest diagonal in a regular octagon?
1. ${\surd }$3 : 1
2. 2 : 1
3. 2 : ${\surd }$3
4. ${\surd }$2 : 1

Choice (D). ${\surd }$2 : 1

Click to watch video solution
Click to view the explanation as a slide show

Detailed Solution

Consider regular octagon ABCDEFGH

Its longest diagonal would be AE or BF or CG or DH.

Let us try to find out AE.

Join AD and draw BP ${\perp }$ AD and CQ ${\perp }$ AD.

PQ = a

AP = QD

a2 = BP2 + AP2 => a2 = 2 AP2 {since BP=AP}

a = ${\surd }$2AP => AP = a/(${\surd }$2)

AD =AP + PQ + QD = a/(${\surd }$2) + a + a/(${\surd }$2)

=>a + a${\surd }$2

AE2 = AD2 + DE2

AE2 = (a + a${\surd }$2) 2 + a2

AE2 = (a2 + 2 x a x ${\surd }$2 + 2a2) + a2

AE2 = a2 (1 + 2${\surd }$2 + 2) + a2

=> a2 (4 + 2${\surd }$2)

Shortest diagonal = AC or CE

AC2 = AB2 + BC2 – 2AB × BC cos135 degree

(Alternatively, we can deduce this using AC2 = AQ2 + QC2. We use cosine rule just to get some practice on a different method.)

= a2 + a2 – 2a2 × ((−1)/(${\surd }$2))

= 2a2 + ${\surd }$2a2

= a2 (2 + ${\surd }$2)

AE2 = a2 (4 + 2${\surd }$2)

"AE2" /"AC2" = "a2 (4 + 2${\surd }$2)" /"a2 (2 + ${\surd }$2)" = 2

"AE" /"AC" = ${\surd }$2

Remember, for a regular octagon.

Each internal angle = 135 degrees.

Each external angle = 45 degrees.

So, we get a bunch of squares and isosceles right–angled ${\triangle }$s if we draw diagonals.

A regular hexagon breaks into equilateral triangles. A regular octagon breaks into isosceles right angled triangles.

Correct Answer: ${\surd }$2 : 1

Our Online Course, Now on Google Playstore!

Fully Functional Course on Mobile

All features of the online course, including the classes, discussion board, quizes and more, on a mobile platform.

Cache Content for Offline Viewing

Download videos onto your mobile so you can learn on the fly, even when the network gets choppy!

More questions from Geometry Triangles

Geometry is probably the most vital topic as far as CAT preparation is concerned. Geometry sets the stage for Trigonometry, Cogeo and Mensuration as well.