the perimeters of two similar triangles are 40 cm and 30 cm respectively. If one side of the first traingle is 21 cm. Determine the corresponding side of the second triangle.

Answer

To solve the problem step by step, we will use the properties of similar triangles, particularly the relationship between the ratios of their corresponding sides and their perimeters.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Perimeter of triangle ABC (P1) = 40 cm
- Perimeter of triangle DEF (P2) = 30 cm
- One side of triangle ABC (AB) = 21 cm
- We need to find the corresponding side of triangle DEF (DE).

2. **Set Up the Ratio of the Perimeters:**
- Since the triangles are similar, the ratio of their corresponding sides is equal to the ratio of their perimeters.
- Therefore, we can write:
\[ \frac{AB}{DE} = \frac{P1}{P2} \]

3. **Substitute the Known Values:**
- Substitute the values of AB, P1, and P2 into the equation:
\[ \frac{21}{DE} = \frac{40}{30} \]

4. **Simplify the Ratio:**
- Simplifying \(\frac{40}{30}\):
\[ \frac{40}{30} = \frac{4}{3} \]
- Now the equation becomes:
\[ \frac{21}{DE} = \frac{4}{3} \]

5. **Cross-Multiply to Solve for DE:**
- Cross-multiplying gives:
\[ 21 \cdot 3 = 4 \cdot DE \]
\[ 63 = 4 \cdot DE \]

6. **Isolate DE:**
- To find DE, divide both sides by 4:
\[ DE = \frac{63}{4} = 15.75 \text{ cm} \]

7. **Final Answer:**
- The corresponding side of the second triangle (DE) is 15.75 cm.