What is the area of the largest equilateral triangle that can be inscribed in a rectangle of axb?

Answer 1

Correction. One vertex of the triangle will coincide the a vertex of the rectangle and the other two vertices will be on the opposite sides of the rectangle (one each of the longer and the shorter side). Use Pythagoras in the three rt triangles thus formed and equate as the sides of the triangle are equal. Thus, find the side.

Answer 2

let a < b
then the largest inscribed equilateral triangle must have side as b we will see how
if a triangle has sides x , y and z, the area of the triangle is
=√[s(s-x)(s-y)(s-z)] where s= (x+y+z)/2
in the case of the equilateral triangle with each side being b
x=y=z=b and s=3b/2, s-x=s-y=s-z=3b/2 -b = b/2
so area =√(3b/2)(b/2)(b/2)(b/2)
=√3b^4/16
=√3b^2/4
since a the equilateral triangle with a as side would have an area =√3a^2/4 which is < √3b^2/4 (area of the equilateral triangle with b as side)
So, if a Conversely if b (We can aslo use pythogoras theorem and get the same answer but that method is a bit longer and a bit more difficult)

Answer 3

Wait, I lied.

The largest triangle that can be put in a rectangle AxB is:

If A < B, Let L be the length of one side of the equilateral triangle

Cos 30 = A/L --> L = A / Cos (30) = A / Sqrt(3)/2 = 2 A / Sqrt(3) = L

So then: L * L/2 = (2 A ^ 2) / 3

(2 A ^ 2) / 3

If I understand your question right, I'm fairly confident this is correct.

Answer 4

In the given rectangle
.Let a be greater than b.
We also notice that the height of the
equilateral triangle can be maximum=b
Let x be the side of the max.
equilateral triangle that can
be inserted in this rectangle.
We have then
x^2=b^2+[x/2]^2
x^2-x^2/4=b^2
x^2=4b^2/3
x=2b/3^1/2
Area of the eq.lat. Triangle
=1/2*base*altitude
=1/2*2/3^1/2*b*b
=b^2/3^1/2

Answer 5

Let a
1. Triangle based on a

Area = 1/2 base x height
= 1/2 a . √(3)a/2
= √(3)a²/4

2. Triangle based on b with height Area = √(3)b²/4 > Triangle in 1 as a
3. Triangle based on b with height a (when b ≥ 2a/ √(3))
Area = 1/2 ab > Triangle in 1 as a
So if a < b and a/b> √3/2 ie √3/2 < a/b < 1
Area = √(3)b²/4

And finally if a< b and a/b ≤ √3/2
Area = 1/2 a * base
and a = √3/2*base so base = 2a/√3
Thus in this case, area = a²/√3 = √(3)a²/3

Answer 6

The formula for area of equilateral triangle is 3/4 * (length of its side)
If a>b,
The area of the largest equilateral triangle will be 3/4 * a.
If b>a,
The area of the largest equilateral triangle will be 3/4 * b.

Answer 7

case:1 if a=b
largest equilateral triangle will be of side "(a)(a)/4"

case:2 if a>b

largest equilateral triangle will be of side "(b)(b)/4"
case:3 if b>a

largest equilateral triangle will be of side "(a)(a)/4"

Answer 8

The triangle will have a measurement of a on all three sides. (assuming a is larger than b)
The triangle will have an area of (1/2a) x a.

Answer 9

if a the equilateraltriangle vertex will be at midpoint of a and have a height b. so if side is x, xCos30=a of x=2a/sqrt(3).
therfore area = sqrt(3)/4*x^2=a^2/sqrt(3)

Answer 10

1/2ab (1/2a>1/2b)