Understanding Similar Triangles and Perimeters

When two triangles are similar, it means their corresponding angles are equal, and their corresponding sides are in proportion. A key property of similar triangles is that the ratio of their perimeters is equal to the ratio of their corresponding sides.

In this problem, we are given that $\Delta$ABC is similar to $\Delta$DEF, denoted as $\Delta$ABC $\sim$ $\Delta$DEF. We are also given the lengths of one pair of corresponding sides, AB and DE, and the perimeter of $\Delta$DEF.

Using the Properties of Similar Triangles

Since $\Delta$ABC $\sim$ $\Delta$DEF, the ratio of corresponding sides is constant:

$$\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}$$

Also, the ratio of their perimeters is equal to the ratio of corresponding sides:

$$\frac{\text{Perimeter of } \Delta\text{ABC}}{\text{Perimeter of } \Delta\text{DEF}} = \frac{AB}{DE}$$

Calculating the Side Ratio

We are given AB = 9.1 cm and DE = 6.5 cm. The ratio of these corresponding sides is:

$$\frac{AB}{DE} = \frac{9.1 \text{ cm}}{6.5 \text{ cm}}$$

To simplify this ratio, we can multiply the numerator and denominator by 10 to remove the decimals:

$$\frac{9.1}{6.5} = \frac{91}{65}$$

Both 91 and 65 are divisible by 13 ($91 = 7 \times 13$, $65 = 5 \times 13$). So, the simplified ratio is:

$$\frac{91}{65} = \frac{7 \times 13}{5 \times 13} = \frac{7}{5}$$

The ratio of corresponding sides is $\frac{7}{5}$.

Finding the Perimeter of ΔABC

Using the property that the ratio of perimeters equals the ratio of corresponding sides:

$$\frac{\text{Perimeter of } \Delta\text{ABC}}{\text{Perimeter of } \Delta\text{DEF}} = \frac{7}{5}$$

We are given that the perimeter of $\Delta$DEF = 25 cm. Substitute this value into the equation:

$$\frac{\text{Perimeter of } \Delta\text{ABC}}{25 \text{ cm}} = \frac{7}{5}$$

Now, solve for the Perimeter of $\Delta$ABC:

$$\text{Perimeter of } \Delta\text{ABC} = \frac{7}{5} \times 25 \text{ cm}$$

$$\text{Perimeter of } \Delta\text{ABC} = 7 \times \frac{25}{5} \text{ cm}$$

$$\text{Perimeter of } \Delta\text{ABC} = 7 \times 5 \text{ cm}$$

$$\text{Perimeter of } \Delta\text{ABC} = 35 \text{ cm}$$

The perimeter of $\Delta$ABC is 35 cm.

Summary of Steps

1. Identify that the triangles are similar.
2. Recall the property that the ratio of perimeters of similar triangles is equal to the ratio of their corresponding sides.
3. Calculate the ratio of the given corresponding sides (AB and DE).
4. Set up the equation using the perimeter ratio and the side ratio.
5. Substitute the known perimeter and solve for the unknown perimeter.

Revision Table: Similar Triangle Properties

Additional Information: Scale Factor and Area Ratio

The ratio of corresponding sides, $\frac{7}{5}$, is also known as the scale factor from $\Delta$DEF to $\Delta$ABC. This means that every length in $\Delta$ABC is $\frac{7}{5}$ times the corresponding length in $\Delta$DEF.

While the ratio of perimeters is equal to the scale factor, the ratio of the areas of similar triangles is equal to the square of the scale factor. In this case, the ratio of the areas would be:

$$\frac{\text{Area of } \Delta\text{ABC}}{\text{Area of } \Delta\text{DEF}} = \left(\frac{7}{5}\right)^2 = \frac{49}{25}$$

Understanding these ratios is crucial for solving various problems involving similar figures.