In △ABC, if a : b : c = 4 : 5 : 6, then the ratio of the circumference to its in radius is
In △ABC, if a : b : c = 4 : 5 : 6, then the ratio of the circumference to its in radius is
16:7
25:11
5:4
9:5
The Correct Option is A
Solution and Explanation
The correct option is: 16:7.
Assuming: a=4k, b=5k, c=6k,
The semi-perimeter, s, is given as: s=2a+b+c=215k.
Hence, the area ΔΔ can be calculated using Heron's formula: Δ=s(s−a)(s−b)(s−c)=15k2⋅7k2⋅5k2⋅3k2=157k2.
As a result, the inradius (r) is determined by: r=sΔ=15k2157k2=7k.
The circumradius (R) can be obtained using the formula: R=4Δabc=4⋅157k24k⋅5k⋅6k=607k120k3=27k.
Thus, the ratio R:r can be expressed as: rR=7k27k=727=2.
Hence, the required ratio is 16:7
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Concepts Used:
Three Dimensional Geometry
Mathematically, Geometry is one of the most important topics. The concepts of Geometry are derived w.r.t. the planes. So, Geometry is divided into three major categories based on its dimensions which are one-dimensional geometry, two-dimensional geometry, and three-dimensional geometry.
Direction Cosines and Direction Ratios of Line:
Consider a line L that is passing through the three-dimensional plane. Now, x,y and z are the axes of the plane and α,β, and γ are the three angles the line makes with these axes. These are commonly known as the direction angles of the plane. So, appropriately, we can say that cosα, cosβ, and cosγ are the direction cosines of the given line L.
