CAT Practice : Geometry-Triangles

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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

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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 = (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

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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.