The dashed triangle is a dilation image of the solid triangle. What is the scale factor?
A coordinate grid shows both the x- and y-axes from -10 to 10 and the graphs of a solid triangle and a dashed triangle.
The solid triangle has vertices at
left parenthesis 4 comma 4 right parenthesis and
left parenthesis 8 comma negative 4 right parenthesis and
left parenthesis negative 4 comma 4 right parenthesis.
The dashed triangle has vertices at
left parenthesis 2 comma 2 right parenthesis and
left parenthesis 4 comma negative 2 right parenthesis and
left parenthesis negative 2 comma 2 right parenthesis.
(1 point)
Responses
one-fourth
Image with alt text: one-fourth
one-half
Image with alt text: one-half
start fraction 2 over 3 end fraction
Image with alt text: start fraction 2 over 3 end fraction
2
2
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To find the scale factor, we can compare the corresponding side lengths of the solid triangle and the dashed triangle.
The first side of the solid triangle has a length of √((8-4)^2 + (-4-4)^2) = √(16+64) = √80.
The corresponding side of the dashed triangle has a length of √((4-2)^2 + (-2-2)^2) = √(4+16) = √20.
The scale factor can be found by dividing the length of the corresponding side of the dashed triangle by the length of the corresponding side of the solid triangle:
Scale factor = √20 / √80
Simplifying this ratio, we have:
Scale factor = √(20/80) = √(1/4) = 1/2
Therefore, the scale factor is one-half.
To find the scale factor, we need to compare the corresponding sides of the solid triangle and the dashed triangle. Let's consider one side of the triangle.
The solid triangle has a side connecting the points (4,4) and (8,-4). The length of this side can be found using the distance formula:
√((x2 - x1)^2 + (y2 - y1)^2)
= √((8 - 4)^2 + (-4 - 4)^2)
= √(4^2 + (-8)^2)
= √(16 + 64)
= √80
= 4√5
Now let's consider the corresponding side of the dashed triangle, which connects the points (2,2) and (4,-2). We can find the length using the distance formula once again:
√((x2 - x1)^2 + (y2 - y1)^2)
= √((4 - 2)^2 + (-2 - 2)^2)
= √(2^2 + (-4)^2)
= √(4 + 16)
= √20
= 2√5
The scale factor is the ratio of the lengths of the corresponding sides:
Scale Factor = Length of corresponding side in dashed triangle / Length of corresponding side in solid triangle
= (2√5) / (4√5)
= 2/4
= 1/2
Therefore, the scale factor is one-half.