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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.
(2 points)
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To determine if the dilation is an enlargement or a reduction, we need to compare the lengths of corresponding sides of the two triangles.
The length of a side is found using the distance formula: √((x2 - x1)^2 + (y2 - y1)^2)
For the solid triangle, the length of the first side is: √((-6 - 6)^2 + (0 - 3)^2) = √((-12)^2 + (-3)^2) = √(144 + 9) = √153
The length of the second side is: √((6 - (-6))^2 + (3 - 6)^2) = √((12)^2 + (-3)^2) = √(144 + 9) = √153
The length of the third side is: √((6 - 6)^2 + (-6 - 3)^2) = √((0)^2 + (-9)^2) = √(0 + 81) = √81 = 9
For the dashed triangle, the length of the first side is: √((-2 - 2)^2 + (0 - 1)^2) = √((-4)^2 + (-1)^2) = √(16 + 1) = √17
The length of the second side is: √((2 - (-2))^2 + (1 - (-2))^2)) = √((4)^2 + (3)^2) = √(16 + 9) = √25 = 5
The length of the third side is: √((2 - 2)^2 + (-2 - 1)^2) = √((0)^2 + (-3)^2) = √(0 + 9) = √9 = 3
To find the scale factor, we can compare the lengths of corresponding sides of the two triangles.
The scale factor for the first side is: (length of dashed triangle / length of solid triangle) = 17 / √153
The scale factor for the second side is: (length of dashed triangle / length of solid triangle) = 5 / √153
The scale factor for the third side is: (length of dashed triangle / length of solid triangle) = 3 / 9 = 1/3
Since all the scale factors are less than 1, the dilation is a reduction.
To determine whether the dilation is an enlargement or a reduction, we can compare the corresponding lengths of the sides of the solid triangle and the dashed triangle.
Let's calculate the lengths of the sides for both triangles:
Solid Triangle:
Side 1 = sqrt((6 - (-6))^2 + (3 - 0)^2) = sqrt(12^2 + 3^2) = sqrt(144 + 9) = sqrt(153)
Side 2 = sqrt((6 - 6)^2 + (-6 - 0)^2) = sqrt(0^2 + (-6)^2) = sqrt(0 + 36) = sqrt(36) = 6
Side 3 = sqrt((6 - (-6))^2 + (-6 - 3)^2) = sqrt(12^2 + (-9)^2) = sqrt(144 + 81) = sqrt(225) = 15
Dashed Triangle:
Side 1 = sqrt((2 - (-2))^2 + (1 - 0)^2) = sqrt(4^2 + 1^2) = sqrt(16 + 1) = sqrt(17)
Side 2 = sqrt((2 - 2)^2 + (-2 - 0)^2) = sqrt(0^2 + (-2)^2) = sqrt(0 + 4) = sqrt(4) = 2
Side 3 = sqrt((2 - (-2))^2 + (-2 - 1)^2) = sqrt(4^2 + (-3)^2) = sqrt(16 + 9) = sqrt(25) = 5
The scale factor of the dilation can be found by comparing the lengths of the corresponding sides of the solid triangle and the dashed triangle:
Scale factor = (Length of side in dashed triangle)/(Length of side in solid triangle)
Scale factor for side 1 = sqrt(17)/sqrt(153) ≈ 0.402
Scale factor for side 2 = 2/6 ≈ 0.333
Scale factor for side 3 = 5/15 = 1/3 ≈ 0.333
Since all the scale factors are less than 1, the dilation is a reduction.