The dashed triangle is a dilation image of the solid triangle with the center at the origin. Is the dilation an enlargement or a reduction? Find the scale factor of the dilation.
A solid triangle has vertices left parenthesis negative 6 comma 0 right parenthesis, left parenthesis 6 comma 3 right parenthesis, and left parenthesis 6 comma negative 6. A dashed triangle has vertices left parenthesis negative 2 comma 0 right parenthesis, left parenthesis 2 comma 1 right parenthesis, and left parenthesis 2 comma negative 2.
To determine whether the dilation is an enlargement or a reduction, we can compare the distances between corresponding points of the solid and dashed triangles.
Let's consider the distance between the first vertex of the solid triangle (-6, 0) and the first vertex of the dashed triangle (-2, 0). The distance between these two points is the absolute value of the difference of their x-coordinates: |-6 - (-2)| = |(-6) + 2| = |(-4)| = 4.
Now let's consider the distance between the second vertices: |6 - 2| = 4.
Finally, let's compare the distance between the third vertices: |(-6) - (-2)| = 4.
Since the distances between corresponding points are all equal, the dilation is an enlargement.
To find the scale factor of the dilation, we can compare the lengths of corresponding sides.
Let's compare the lengths of the sides formed by the first and second vertices: |(-6 - 6)|/|(-2 - 2)| = 12/4 = 3.
Now let's compare the lengths of the sides formed by the second and third vertices: |(6 - 6)|/|(2 - 2)| = 0/0 = undefined.
Since not all corresponding side lengths are equal, we cannot determine a single scale factor for the dilation. The scaling appears to be different for different sides.
To determine if the dilation is an enlargement or a reduction, we can compare the lengths of corresponding sides.
Let's start by calculating the scale factor. The scale factor is defined as the ratio of the length of a side of the dashed triangle to the length of the corresponding side of the solid triangle.
Using the distance formula, we can find the lengths of the sides:
For the solid triangle:
Side A: √[(-6 - 6)^2 + (0 - 3)^2] = √144 + 9 = √153
Side B: √[(6 - (-6))^2 + (3 - 0)^2] = √144 + 9 = √153
Side C: √[(6 - 6)^2 + (-6 - 3)^2] = √0 + 81 = √81 = 9
For the dashed triangle:
Side A': √[(-2 - 2)^2 + (0 - 1)^2] = √16 + 1 = √17
Side B': √[(2 - (-2))^2 + (1 - 0)^2] = √16 + 1 = √17
Side C': √[(2 - 2)^2 + (-2 - 1)^2] = √0 + 9 = √9 = 3
Now, let's compare the lengths of the corresponding sides:
Side A' / Side A = √17 / √153 ≈ 0.415
Side B' / Side B = √17 / √153 ≈ 0.415
Side C' / Side C = 3 / 9 = 1/3
Since all the scale factors are less than 1, the lengths of the corresponding sides have reduced. Therefore, the dilation is a reduction.
The scale factor of the dilation is approximately 0.415.
To determine if the dilation is an enlargement or a reduction, we can compare the lengths of corresponding sides of the solid triangle and the dashed triangle. If the corresponding sides are larger in the dashed triangle, then it is an enlargement. If the corresponding sides are smaller, then it is a reduction.
Let's calculate the scale factor by comparing the lengths of corresponding sides.
Solid triangle:
Side A: Length = √[(6 - (-6))^2 + (3 - 0)^2] = √(12^2 + 3^2) = √(144 + 9) = √153
Dashed triangle:
Side A': Length = √[(2 - (-2))^2 + (1 - 0)^2] = √(4^2 + 1^2) = √(16 + 1) = √17
The scale factor is given by the ratio of the lengths of corresponding sides: Scale factor = Length of A' / Length of A = √17 / √153
To simplify the scale factor, we can multiply the numerator and denominator by √153 to eliminate the square roots:
Scale factor = (√17 / √153) * (√153 / √153) = (√(17 * 153) / √(153 * 153)) = √(2601 / 23409)
So, the scale factor of the dilation is √(2601 / 23409).
Since the length of side A' is smaller than the length of side A, we can conclude that the dilation is a reduction.