How do I do this:

Question

591. Find the side of the largest square that can be drawn inside a 12-inch equilateral triangle,

one side of the square aligned with one side of the triangle.

Answer 1

height of an equilateral triangle with side = 12 inches is 6sqr3 inches

graph the triangle with middle of the base at the origin, apex = (0,6sqr3) right vertex = (6,0)

center of the base of the inscribed square is the origin as well.

equation of the line of the right side of the triangle is: y= (-sqr3)x + 6sqr3

set y = 2x = (sqr3)x + 6sqr3

(2-sqr3)x = 6sqr3

x =6sqr3/(2+sqr3) = (12sqr3 -18)/(4-3)

= 12sqr3 -18= about 2.8

2x = about 5.6

the inscribed square with maximum area is about 5.6 by 5.6 inches

or exactly 12sqr3 -18 by 12sqr3-18 inches

with max area = about 31 in^2