We are given two trapezoids, $ABCD$ and $WXYZ$, which are similar (denoted $ABCD \sim WXYZ$). Our task is to find the value of $p$, which represents the length of side $CD$ in trapezoid $ABCD$.

Similarity of Trapezoids:

Since the trapezoids are similar, the corresponding sides are proportional. This gives us the relationship:

$\frac{AB}{WX} = \frac{BC}{XY} = \frac{CD}{YZ} = \frac{DA}{ZW}$

Given Measurements:

From the image:

• In trapezoid $WXYZ$:

  • $WX = 7 \, \text{mm}$
  • $XY = 13 \, \text{mm}$
  • $YZ = 10 \, \text{mm}$
  • $ZW = 13 \, \text{mm}$

• In trapezoid $ABCD$:

  • $AB = 13.3 \, \text{mm}$
  • $BC = 24.7 \, \text{mm}$
  • $DA = 24.7 \, \text{mm}$
  • $CD = p \, \text{mm}$ (unknown)

Find the Scaling Ratio:

We can find the scale factor using corresponding sides. Let's use the ratio of $AB$ to $WX$:

$\frac{AB}{WX} = \frac{13.3}{7} \approx 1.9$

This means every side in trapezoid $ABCD$ is approximately 1.9 times longer than the corresponding side in trapezoid $WXYZ$.

Solve for $p$:

Using the ratio for the sides $CD$ and $YZ$:

$\frac{CD}{YZ} = \frac{AB}{WX} = 1.9$

So:

$p = 1.9 \times YZ = 1.9 \times 10 = 19 \, \text{mm}$

Thus, the value of $p$ is $19 \, \text{mm}$.

Would you like a further breakdown or have any questions?

Related Questions:

1. What other properties of similar figures can help in solving problems like this?
2. How does the ratio of areas relate to the ratio of corresponding sides in similar figures?
3. Can you solve for other unknowns if trapezoids were not similar?
4. What happens to the ratio if the figures are scaled differently along different axes?
5. How can trapezoid properties (like the height) be used in solving for area?

Tip:

Always check the scale factor using multiple pairs of corresponding sides to ensure accuracy when working with similar figures.