To solve the problem:

Given:

• Triangle HIJ is dilated by a scale factor of $\frac{1}{2}$ to form triangle $H'I'J'$.
• The side $H'I'$ of the image triangle measures 12.

To find:

The measure of side $HI$ in the original triangle.

Solution:

In a dilation, the original side length is divided by the scale factor to obtain the image's side length. Here, the reverse is true: to find the original side length, multiply the image side length by the reciprocal of the scale factor.

$HI = H'I' \times \text{(reciprocal of the scale factor)} = 12 \times 2 = 24$

Answer:

The measure of side $HI$ is $24$.

Would you like further clarification or details?

Related Questions:

1. How does a scale factor affect all side lengths of a triangle during dilation?
2. What happens to the area of a triangle when it is dilated with a scale factor of $\frac{1}{2}$?
3. How do you determine the scale factor if given the original and image side lengths?
4. What properties remain unchanged during dilation of a triangle?
5. Can dilation produce congruent triangles? Why or why not?

Tip:

Always multiply by the reciprocal of the scale factor when finding original dimensions from dilated ones.