CBSE Class 12-science Answered
Prove that the triangle of maximum area that can be inscribed in a circle is an equilateral triangle
Asked by agarrajeev06 | 20 Jan, 2019, 12:46: PM
Expert Answer
Let us draw a circle of radius R in a cartesian coordinate system , with centre (0,R) as shown in figure.
Let us inscribe a triangle ABC, so that side BC is parallel to x-axis and vertex A is at y-axis with coordinate A(0,2R)
Area of circle Δ = (1/2)(BC)(AD) = (1/2) (2x) (2R-y) = x (2R-y) ...................(1)
eqn. of circle : x2 +(y - R)2 = R2 or x2 = 2yR - y2 ....................(2)
hence using eqn.(2), eqn.(1) can be written as : 
.....................(3)
.....................(3)To get extreme values of Δ, with respect to y, we differentiate eqn.(3) and equate to zero
..............................(4)we get from (4), extreme values are at y = (R/2) or y = 2R
when y = 2R, area of triangle is zero and minimum
when y = (R/2), area of triangle is maximum
if y =(R/2), then from eqn.(2) we get, x= (√3/2)R
side BC = 2x = √3 R ; BD = (√3/2)R , AD = 2R-y = (3/2)R , hence it can be shown that AB = AC = √3 R
hence BC = AB = AC , or triangle ABC is equilateral
Answered by Thiyagarajan K | 20 Jan, 2019, 04:16: PM
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