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
To determine the scale factor of a dilation, you can compare the lengths of corresponding sides in the two triangles.
Let's consider the solid triangle first. The distance between two of its vertices, say (4, 4) and (8, -4), can be calculated using the distance formula for points in a coordinate plane:
Distance = √((x2 - x1)² + (y2 - y1)²)
For the solid triangle, let's calculate the distance between the vertices (4, 4) and (8, -4):
Distance_solid = √((8 - 4)² + (-4 - 4)²) = √((4)² + (-8)²) = √(16 + 64) = √80
Now let's calculate the distance between the corresponding vertices (2, 2) and (4, -2) in the dashed triangle:
Distance_dashed = √((4 - 2)² + (-2 - 2)²) = √((2)² + (-4)²) = √(4 + 16) = √20
To find the scale factor, we divide the length of a side of the dashed triangle by the corresponding length of a side in the solid triangle.
Scale factor = Distance_dashed / Distance_solid
Scale factor = √20 / √80
Scale factor = √(20/80)
Scale factor = √(1/4)
Scale factor = 1/2
Therefore, the scale factor of the dilation from the solid triangle to the dashed triangle is one-half.