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Given a pair of similar polygons, solve for *x*.

We are given two similar trapezoids: \(\triangle PQRS\) and \(\triangle ABCD\). The sides of these trapezoids are proportional, and we are tasked with finding \(x\), the missing side length in \(\triangle ABCD\).

Since the trapezoids are similar, the ratio of corresponding sides must be equal. Let's set up the ratios using the corresponding sides:

- \(PQ\) corresponds to \(AB\),
- \(QR\) corresponds to \(BC\),
- \(RS\) corresponds to \(CD\),
- \(SP\) corresponds to \(DA\).

The known values are:
- \(PQ = 35 \, \text{cm}, AB = x \, \text{cm}\),
- \(QR = 25 \, \text{cm}, BC = 15 \, \text{cm}\),
- \(RS = 30 \, \text{cm}, CD = 18 \, \text{cm}\),
- \(SP = 20 \, \text{cm}, DA = 12 \, \text{cm}\).

We can use the ratio of corresponding sides to find \(x\). Taking the ratio of \(PQ\) to \(AB\), we have:
\[
\frac{PQ}{AB} = \frac{QR}{BC} = \frac{RS}{CD} = \frac{SP}{DA}.
\]

Using the ratio \(\frac{QR}{BC}\):
\[
\frac{25}{15} = \frac{35}{x}.
\]

Now, let's solve for \(x\).The value of \(x\), which corresponds to the length of side \(AB\), is \(21 \, \text{cm}\).

Would you like more details or have any further questions?

Here are 5 related questions for further practice:
1. How do you prove that two polygons are similar?
2. If the ratio of two corresponding sides is 2:3, how does it affect the areas of similar polygons?
3. How would the angles in two similar polygons compare?
4. Can two trapezoids with the same area be non-similar? Why or why not?
5. If another side was missing in the larger trapezoid, how would you calculate it?

**Tip:** In similar polygons, corresponding angles are always equal, and the sides are proportional.

Given a pair of similar polygons, solve for x.

Solve for the missing side x in a pair of similar trapezoids using proportionality of corresponding sides. Learn key concepts in geometry and similar polygons, along with ratios and proportions. Suitable for students in Grades 8-10.

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