A dilation maps Upper DeltaUpper TUpper UUpper V onto Upper DeltaUpper T primeUpper U primeUpper V prime. Find the missing values.

Question

Upper TUpper Uequals2 in
Upper T primeUpper U primeequals6 in
Upper UUpper Vequals9 in
Upper U primeUpper V primeequals​__ in
Upper TUpper Vequals8 in
Upper T primeUpper V primeequals​__ in

Question content area bottom
Part 1
Upper TUpper Uequals2 in
Upper T primeUpper U primeequals6 in
Upper UUpper Vequals9 in
Upper U primeUpper V primeequals
  
enter your response here in
Upper TUpper Vequals8 in
Upper T primeUpper V primeequals
  
enter your response here in

Answer 1

In a dilation, the segments are proportional. The relationship can be expressed as:

\[ \frac{\text{Length of an original segment}}{\text{Corresponding length in the image}} = \text{Scale Factor} \]

Let's denote the lengths of the segments in triangle \( TUV \) as follows:

• \( TU = 2 \) in

• \( T'U' = 6 \) in

• \( UV = 9 \) in

• \( U'V' = x \) in (missing)

• \( TV = 8 \) in

• \( T'V' = y \) in (missing)

We can start by calculating the scale factor using the segments that we know:

\[ \text{Scale Factor} = \frac{T'U'}{TU} = \frac{6}{2} = 3 \]

Now we can use this scale factor to find the missing lengths:

For \( U'V' \):

Using the scale factor on segment \( UV \):

\[ \frac{U'V'}{UV} = \text{Scale Factor} \]

\[ \frac{x}{9} = 3 \]

Now solve for \( x \):

\[ x = 3 \times 9 = 27 \]

Thus, \( U'V' = 27 \) in.

For \( T'V' \):

Similarly, using the scale factor on segment \( TV \):

\[ \frac{T'V'}{TV} = \text{Scale Factor} \]

\[ \frac{y}{8} = 3 \]

Now solve for \( y \):

\[ y = 3 \times 8 = 24 \]

Thus, \( T'V' = 24 \) in.

Summary of Results:

• \( U'V' = 27 \) in
• \( T'V' = 24 \) in

You can fill in the blanks with these values:

1. \( U'V' = 27 \) in
2. \( T'V' = 24 \) in