Question

Triangle TUV was dilated to create triangle T'U'V' using point A as the center of dilation.

Point A is the center of dilation. Triangle T U V is dilated to create smaller triangle T prime U prime V prime. The length of A V prime is 3.2. The length of V prime V is 4.8.

What is the scale factor of the dilation?

Two-fifths
Two-thirds
Three-halves
Eight-fifths

Answer 1

To find the scale factor of the dilation from triangle TUV to triangle T'U'V', we can use the lengths provided.

The length of AV' (which is the distance from point A to vertex V') is given as 3.2, and the length of VV' (which is the distance from vertex V to vertex V') is given as 4.8.

In a dilation, the relationship between the distances is expressed as follows:

\[
\text{scale factor} = \frac{\text{length from center to image}}{\text{length from center to pre-image}}
\]

In this case,
- Length from A to V' (image) = 3.2
- Length from A to V (pre-image): Since AV = AV' + VV', we can calculate:

\[
AV = AV' + VV' = 3.2 + 4.8 = 8.0
\]

Now we can now find the scale factor (k):

\[
k = \frac{AV'}{AV} = \frac{3.2}{8.0}
\]

To simplify \( \frac{3.2}{8.0} \):

\[
k = \frac{3.2 \div 3.2}{8.0 \div 3.2} = \frac{1}{2.5} = \frac{2}{5}
\]

Thus, the scale factor of the dilation is **two-fifths**.