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. 3 : 1
    2. 2 : 1
    3. 2 : 3
    4. 2 : 1

 

  • Correct Answer
    Choice (D). 2 : 1

Explanatory Answer

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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 AD and CQ AD.

PQ = a

AP = QD

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

a = 2AP => AP = a/(2)

AD =AP + PQ + QD = a/(2) + a + a/(2)

=>a + a2

AE2 = AD2 + DE2

AE2 = (a + a2) 2 + a2

AE2 = (a2 + 2 x a x 2 + 2a2) + a2

AE2 = a2 (1 + 22 + 2) + a2

=> a2 (4 + 22)

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)/(2))

= 2a2 + 2a2

= a2 (2 + 2)

AE2 = a2 (4 + 22)

"AE2" /"AC2" = "a2 (4 + 22)" /"a2 (2 + 2)" = 2

"AE" /"AC" = 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 s if we draw diagonals.

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

Answer choice (d)

Correct Answer: 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.