Question
Mathematics Question on Three Dimensional Geometry
In △ABC, if a : b : c = 4 : 5 : 6, then the ratio of the circumference to its in radius is
A
16:7
B
25:11
C
5:4
D
9:5
Answer
16:7
Explanation
Solution
The correct option is: 16:7.
Assuming: a =4 k , b =5 k , c =6 k ,
The semi-perimeter, s , is given as: s =2 a +b +c =215 k .
Hence, the area ΔΔ can be calculated using Heron's formula: Δ=s(s − a)(s − b)(s − c)=15 k 2⋅7 k 2⋅5 k 2⋅3 k 2=157 k 2.
As a result, the inradius (r) is determined by: r =s Δ=15 k 2157 k 2=7 k.
The circumradius (R) can be obtained using the formula: R =4Δ abc =4⋅157 k 24 k ⋅5 k ⋅6 k =607 k 120 k 3=27 k.
Thus, the ratio R :r can be expressed as: rR =7 k 27 k =727=2.
Hence, the required ratio is 16:7