Show that the triangle of maximum area that can be inscribed in a given
circle is an equilateral triangle.
Answers
Proved triangle of maximum area that can be inscribed in a given circle is an equilateral triangle.
Step-by-step explanation:
Refer attached picture:
R = Radius of circle
2r = Base of Triangle
h = altitude of triangle
BD = CD = 2r/2 = r
in Δ OBD
R² = r² + (h - R)²
=> r² = 2hR - h²
Area of Traingle = (1/2) * Base * Altitude
= (1/2) * 2r * h
= r * h
Let say Z = Area²
=> Z = r²h²
=> Z = (2hR - h²)h²
=> Z = 2h³R - h⁴
dZ/dh = 6h²R - 4h³
6h²R - 4h³ = 0
=> 3R - 2h = 0
=> h = 3R/2
d²Z/dr² = 12hR - 12h²
at h = 3R/2
= 18R² - 27R²
= -9R² (-ve)
=> h = 3R/2 will give maximum Area
r² = 2hR - h²
=> r² = 3R² - 9R²/4
=> r² = 3R²/4
=> r = √3 R/2
=> 2r = √3 R
=> BC = √3 R
AB² = AC² = h² + r² = 9R²/4 + 3R²/4 = 3R²
=> AB = AC = √3R
AB = AC = BC = √3 R
=> Equilateral Triangle
Hence triangle of maximum area that can be inscribed in a given circle is an equilateral triangle.
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