There is a picture of a square - BCAD, and then there is a triangle EAD.
B. E C
A. D
FIGURE NOT DRAWN TO SCALE
In the figure above, the perimeter of the equilateral triangle AED IS 6. What is the area of the rectangle ABCD?
A) 6
B) 4
C) 4 radical 3
D) 3 radical 3
E) 2 radical 3
wondering where E is.
clearly the base of ∆EAD is AD.
So, each of its sides has length 2.
Thus AD=2, and the area of ABCD is 2^2 = 4
Not sure what the EC is, but if you want the length, then since the altitude of ∆EAD is √3,
1^2 + (2+√3)^2 = EC^2
EC = √2+√6
unless, of course, E is inside ABCD. In that case,
1^2 + (2-√3)^2 = EC^2
EC = √6-√2
Why did you do
1^2 + ( 2+ radical3) ^2
Is that the Pythagorean theorem?
If it is why did you do
( 2+ radical3) ^2
Wouldn't it be
(Radical 3)^2 + EC = 2^2
Maybe. I figured EC was the hypotenuse of
∆ECF where F is the midpoint of BC -- the side parallel to AD.
If I got it wrong, then you may be right.
Though, I doubt it, since that makes EC=1.
To find the area of the rectangle ABCD, we first need to find the length and width of the rectangle.
Given that the perimeter of the equilateral triangle AED is 6, we know that each side of the triangle has a length of 2.
Since the triangle is equilateral, we can use the properties of equilateral triangles to determine the length of the rectangle. The length of the rectangle is equal to the height of the equilateral triangle.
The height of an equilateral triangle can be found by using the formula:
height = (side length * sqrt(3)) / 2
In this case, the side length is 2. Plugging this into the formula, we have:
height = (2 * sqrt(3)) / 2
= sqrt(3)
So, the height of the rectangle ABCD is sqrt(3).
Since the rectangle is formed by the bases AD and BC of the equilateral triangle, the width of the rectangle is equal to the side length of the equilateral triangle.
Therefore, the width of the rectangle ABCD is 2.
Finally, we can calculate the area of the rectangle by multiplying the length and width:
Area = Length * Width
= sqrt(3) * 2
= 2 sqrt(3)
So, the area of the rectangle ABCD is 2 sqrt(3).
Therefore, the correct answer is C) 4 radical 3.