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

Question

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

set the triangle with the base on the x axis

half the triangle in quadrant I and half in quadrant II

each angle = 60 degrees = pi/3

it helps to do a rough sketch

tan(pi/3) = sqr3 = height of the triangle divided by half its base = (6sqr3)/6 = sqr3 = 2x/(6-x)

2x = 6sqr3 - xsqr3

(2+sqr3)x = 6sqr3

x = 6sqr3/(2+sqr3)

side of the square of maximum area = 2x = 12sqr3/(2+sqr3) inches = about 5.57 inches

(side note, almost irrelevant. Although a square has the largest area for any rectangle of given perimeter, the largest area for a rectangle inscribed in an equilateral triangle is not a square. The square, about 5.57 inches has area = 5.57^2 = about 31.02 square inches. the largest inscribed rectangle is 6 by 3sqr3 or about 6 by 5.2 inches with area = 18sqr3 = about 31.18 square inches, It's almost a square , but not quite)

Answer 2

Let's first sketch a diagram so we can visualize the problem. Draw a right side up equilateral triangle and label the vertices A, B, and C, where A is the top vertex and B and C are the bottom left and bottom right vertices, respectively. Then draw an inscribed square DEFG, where D and E are the bottom left and bottom right vertices and F and G are the upper right and upper left vertices. DE should be centered on BC and points F and G should be touching AC and AB, respectively.

We are looking for the length of the square, so that's our unknown. So let's label DG with length x. Since we are given that the triangle has sides of length 12 inches, we can deduce that BD must have length 6-x/2. We also know that triangle BDG is a 30-60-90 triangle, so the ratio of the two legs of that triangle is √3. Thus, we can create an equation for us to solve:

x = √3(6-x/2)

x = 6√3 - x√3/2

x + x√3/2 = 6√3

x = 6√3 / (1 + √3/2)

Multiplying the top and bottom of the right side by the conjugate and simplifying, we get:

x = 24√3 - 36

x ≈ 5.569

So the square has an approximate side length of 5.569 inches.