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.
I think it’s a reduction because the dashed triangle is smaller than the other triangle, I just don’t know how to explain it. Is my reasoning correct? And how do I prove it?
Yes, your reasoning is correct. If the dashed triangle is smaller than the solid triangle, it means the image is reduced.
To prove it mathematically, you can compare the distances between corresponding points on the two triangles, specifically the distance from the origin to any point on the solid triangle and its corresponding point on the dashed triangle.
Let's say the distance from the origin to a point on the solid triangle is d1, and the corresponding distance on the dashed triangle is d2. Since the dashed triangle is smaller, d2 is less than d1.
Now, let's find the scale factor of the dilation. The scale factor is the ratio of the lengths of corresponding sides of the two triangles. Let's take the length of a side of the solid triangle as s and the corresponding side of the dashed triangle as s'.
Since the dashed triangle is a reduction, the scale factor is less than 1. You can find the scale factor by dividing the length of a side of the dashed triangle by the length of the corresponding side of the solid triangle: s'/s.
In summary, the dilation is a reduction because the dashed triangle is smaller than the solid triangle, and the scale factor is less than 1 because the lengths of corresponding sides of the two triangles have a ratio less than 1.