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The problem asks us to find the area of triangle \( X'Y'Z' \) after a dilation by a scale factor of \( \frac{1}{2} \) from the original triangle \( XYZ \), whose area is given as 20 square inches.

### Understanding the Problem:
1. **Dilation and Area**: When a shape is dilated by a scale factor \( k \), the area of the new shape is multiplied by \( k^2 \). In this case, the scale factor \( k \) is \( \frac{1}{2} \).

2. **Calculate the Area of the Dilated Triangle**:
   \[
   \text{New Area} = \text{Original Area} \times \left(\frac{1}{2}\right)^2
   \]
   Substituting the values:
   \[
   \text{New Area} = 20 \times \left(\frac{1}{2}\right)^2 = 20 \times \frac{1}{4} = 5 \text{ square inches}
   \]

### Answer:
The area of triangle \( X'Y'Z' \) is \( \boxed{5} \) square inches.

Would you like more details on the dilation process or have any other questions?

Here are 5 related questions to expand on this topic:

1. What happens to the perimeter of a triangle when it is dilated by a scale factor of \( \frac{1}{2} \)?
2. How would the area change if the scale factor was \( \frac{1}{3} \) instead of \( \frac{1}{2} \)?
3. How does dilation affect other geometric properties, such as angles or side lengths?
4. What would the area of the triangle be if it was dilated by a scale factor of 2 instead of \( \frac{1}{2} \)?
5. How does the scale factor affect volume if this was a 3D object instead of a 2D triangle?

**Tip**: Remember, for any dilation, the area changes by the square of the scale factor, and the volume (for 3D objects) changes by the cube of the scale factor.

Triangle XYZ is dilated by a scale factor of 1/2 to create triangle X'Y'Z'. If the area of triangle XYZ is 20 square inches, what is the area of triangle X'Y'Z'?

Learn how to calculate the area of a triangle after a dilation by a scale factor of 1/2. In this example, triangle XYZ has an initial area of 20 square inches. After dilation, the area of triangle X'Y'Z' is calculated using the formula for area scaling. Suitable for students in Grades 7-9.

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