A right triangle is drawn in a semicircle of radius (1/2) with one of its legs along the diameter. The maximum area of the triangle is

Question

A right triangle is drawn in a semicircle of radius (1/2) with one of its legs along the diameter. The maximum area of the triangle is (A) (1/4) (B) (3 √3/32) (C) (3 √3/16) (D) (1/8)

Tardigrade Admin · Accepted Answer

( BC )( CE )= x (1- x ) ( text property of circle) [ AC = x ; CD =1- x ] but BC = CE ∴ BC =√ x (1- x ) A =( x √ x - x 2/2) text . Now maximise A . A max occurs at x =(3/4)

Q. A right triangle is drawn in a semicircle of radius $\frac{1}{2}$ with one of its legs along the diameter. The maximum area of the triangle is

$\frac{1}{4}$

$\frac{3 3}{32}$

$\frac{3 3}{16}$

$\frac{1}{8}$

$(BC) (CE) = x (1 - x) (property of circle) [A C = x; C D = 1 - x]$
but $BC = CE$
$∴ BC = x (1 - x)$
$A = \frac{x x - x ^{2}}{2} . Now maximise A .$
$A_{m a x}$ occurs at $x = \frac{3}{4}$

Q. A right triangle is drawn in a semicircle of radius 21​ with one of its legs along the diameter. The maximum area of the triangle is

41​

3233​​

1633​​

81​