Question

Parallelogram P was dilated to form parallelogram P prime.

Parallelogram P has side lengths of 10 and x + 3. Parallelogram P prime has side lengths of 20 and 8.

What is the value of x? (Figures not drawn to scale.)

Answer 1

To solve for the value of \( x \), we need to analyze the dilations of the side lengths of parallelograms \( P \) and \( P' \).

Given the side lengths of parallelogram \( P \):
- One side length is \( 10 \)
- The other side length is \( x + 3 \)

Given the side lengths of parallelogram \( P' \):
- One side length is \( 20 \)
- The other side length is \( 8 \)

Since \( P \) is dilated to form \( P' \), the sides of \( P' \) must correspond to the sides of \( P \) with a constant dilation ratio.

Let's determine the dilation ratio between the sides:

1. For the side lengths \( 10 \) (from \( P \)) and \( 20 \) (from \( P' \)), we can find the scale factor:
\[
\text{Scale factor} = \frac{\text{Length of } P'}{\text{Length of } P} = \frac{20}{10} = 2
\]

2. Next, we apply the same scale factor to the side length \( x + 3 \):
\[
\text{Length of } P' = 2 \times \text{Length of } P
\]
This gives us:
\[
8 = 2(x + 3)
\]

Now we can solve for \( x \):
\[
8 = 2(x + 3)
\]
Dividing both sides by 2:
\[
4 = x + 3
\]
Subtracting 3 from both sides:
\[
x = 4 - 3 = 1
\]

Thus, the value of \( x \) is:
\[
\boxed{1}
\]