Geometry - Diagonals of octagon

Given an octagon, we need to find the ratio of longest diagonal to the shortest diagonal in the regular octagon.

1. Geometry Q11

2. Qn : Diagonals of octagon (a)  3 : 1 (b) 2 : 1 (c) 2 :  3 (d)  2 : 1 What is the ratio of longest diagonal to the shortest diagonal in a regular octagon?

3. Soln : Diagonals of octagon Consider regular octagon ABCDEFGH What is the ratio of longest diagonal to the shortest diagonal in a regular octagon? B A C E D F H G P Q a a a a a a a a

4. Soln : Diagonals of octagon 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 a 2 = BP 2 + AP 2  a 2 = 2 AP 2 {since BP=AP} a =  2AP  AP = AD =AP + PQ + QD = + a +   What is the ratio of longest diagonal to the shortest diagonal in a regular octagon?

5. Soln : Diagonals of octagon a + a  2 AE 2 = AD 2 + DE 2 AE 2 = ( a + a  2) 2 + a 2 AE 2 = ( a 2 + 2 x a x 2  2 + 2a 2 ) + a 2 AE 2 = a 2 (1 + 2  2 + 2) + a 2  a 2 (4 + 2  2) Shortest diagonal = AC or CE AC 2 = AB 2 + BC 2 – 2AB × BC cos135  What is the ratio of longest diagonal to the shortest diagonal in a regular octagon?

6. Soln : Diagonals of octagon (Alternatively, we can deduce this using AC 2 = AQ 2 + QC 2 . We use cosine rule just to get some practice on a different method .) = a 2 + a 2 – 2a 2 × ( ) = 2a 2 +  2a 2 = a 2 (2 +  2) AE 2 = a 2 (4 + 2  2 ) = = 2   What is the ratio of longest diagonal to the shortest diagonal in a regular octagon?

7. Soln : Diagonals of octagon =  2 Remember, for a regular octagon. Each internal angle = 135  Each external angle = 45  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)   What is the ratio of longest diagonal to the shortest diagonal in a regular octagon?