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We are given that triangles ABC and PQR are mathematically similar, meaning their corresponding sides are proportional. We need to find the value of \( RP = x \).

Let's label the corresponding sides:
- \( AB \) corresponds to \( PQ \)
- \( AC \) corresponds to \( PR \)
- \( BC \) corresponds to \( QR \)

From the figure:
- \( AC = 1.61 \, \text{cm} \)
- \( BC = 2.8 \, \text{cm} \)
- \( QR = 3.2 \, \text{cm} \)
- We are tasked with finding \( RP = x \, \text{cm} \).

Since the triangles are similar, the ratio of corresponding sides is equal. We can set up a proportion using \( AC \) and \( PR \):

\[
\frac{AC}{PR} = \frac{BC}{QR}
\]

Substituting the values:

\[
\frac{1.61}{x} = \frac{2.8}{3.2}
\]

Now, we solve for \( x \):

\[
\frac{2.8}{3.2} = 0.875
\]

Thus:

\[
\frac{1.61}{x} = 0.875
\]

Multiplying both sides by \( x \) and then dividing by 0.875:

\[
1.61 = 0.875x
\]
\[
x = \frac{1.61}{0.875}
\]

Let's calculate the value of \( x \).The value of \( RP = x \) is \( 1.84 \, \text{cm} \).

Would you like further details on the solution or have any questions?

Here are some related questions for practice:
1. If the perimeter of triangle ABC is 6.21 cm, what is the perimeter of triangle PQR?
2. How do you determine if two triangles are similar based on their angles?
3. If triangle ABC has an area of 2 cm², what is the area of triangle PQR?
4. What is the ratio of the area of two similar triangles based on their side lengths?
5. If you only knew the ratio of one pair of corresponding sides, how could you find the others?

**Tip:** For similar triangles, the ratio of any two corresponding sides remains constant across the triangles.

Triangle ABC is mathematically similar to triangle PQR. Find the value of x.

This example demonstrates how to find the value of a missing side in two mathematically similar triangles using the proportionality of their corresponding sides. The problem uses a step-by-step approach to solve for x, the side RP in triangle PQR. Suitable for students in Grades 8-10.

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