Let Q be the center of equilateral triangle ABC. A dilation centered at Q with scale factor -4/3 is applied to triangle ABC, to obtain triangle A'B'C'. Let S be the area of the region that is contained in both triangles ABC and A'B'C'. Find A/[ABC].
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To find the area ratio A/[ABC], we need to calculate the area of triangle A'B'C' and triangle ABC.
Let's first calculate the area of triangle ABC.
Since ABC is an equilateral triangle, we know that all sides are equal and all angles are 60 degrees.
Let's assume the side of the equilateral triangle ABC is "s".
The area of any equilateral triangle can be calculated using the formula: Area = (s²√3) / 4
Therefore, the area of triangle ABC will be: A[ABC] = (s²√3) / 4
Now, let's find the area of triangle A'B'C':
Since a dilation is applied with a scale factor of -4/3 centered at Q, the side lengths of triangle A'B'C' will be (4/3) times the side lengths of triangle ABC.
Let's assume the side length of triangle A'B'C' is "x".
Then the side length of triangle ABC will be (3/4)x.
Now, let's find the area of triangle A'B'C':
The area of any equilateral triangle can be calculated using the formula: Area = (x²√3) / 4
Therefore, the area of triangle A'B'C' will be: A[A'B'C'] = ((3/4)x)²√3 / 4
Finally, to find the area of the region that is contained in both triangles ABC and A'B'C' (S), we need to find the area of the intersecting region.
Since both triangles are identical, the intersecting region will also be an equilateral triangle.
To find the area of the intersecting region, we need to calculate the side length of the intersecting equilateral triangle.
The scale factor for the dilation is -4/3, so the side length of the intersecting triangle will be (4/3) times the side length of the intersecting region in triangle ABC.
Therefore, the side length of the intersecting equilateral triangle will be (4/3) * ((3/4)x).
Now, let's calculate the area of the intersecting triangle:
Area of the intersecting triangle = ((4/3) * ((3/4)x))²√3 / 4
Finally, the area S that is contained in both triangles ABC and A'B'C' will be the area of the intersecting triangle:
S = ((4/3) * ((3/4)x))²√3 / 4
Now, to find the ratio A/[ABC], we can divide A[A'B'C'] (the area of triangle A'B'C') by A[ABC] (the area of triangle ABC):
A/[ABC] = A[A'B'C'] / A[ABC]
= (((3/4)x)²√3/4) / ((s²√3) / 4)
= ((9/16)x²√3) / (s²√3)
= (9/16)(x²/s²)
So, the ratio A/[ABC] is (9/16)(x²/s²). This is the final answer.