Understanding Similar Triangles and Perimeters

The problem involves two similar triangles, Δ ABC and Δ DEF. We are given their perimeters and asked to find the ratio of the sum of two sides in one triangle to the sum of the corresponding two sides in the other triangle.

Key Concept: Properties of Similar Triangles

When two triangles are similar, the ratio of their corresponding sides is constant. This constant ratio is also equal to the ratio of their perimeters.

Given that Δ ABC is similar to Δ DEF, we can write the ratio of corresponding sides:

\[ \frac{\text{AB}}{\text{DE}} = \frac{\text{BC}}{\text{EF}} = \frac{\text{CA}}{\text{FD}} \]

Also, the ratio of their perimeters is equal to this same constant ratio:

\[ \frac{\text{Perimeter of } \Delta \text{ABC}}{\text{Perimeter of } \Delta \text{DEF}} = \frac{\text{AB} + \text{BC} + \text{CA}}{\text{DE} + \text{EF} + \text{FD}} = \frac{\text{AB}}{\text{DE}} = \frac{\text{BC}}{\text{EF}} = \frac{\text{CA}}{\text{FD}} \]

Calculating the Ratio of Perimeters

We are given the perimeters:

• Perimeter of Δ ABC = 40 cm
• Perimeter of Δ DEF = 30 cm

The ratio of their perimeters is:

\[ \frac{\text{Perimeter of } \Delta \text{ABC}}{\text{Perimeter of } \Delta \text{DEF}} = \frac{40 \text{ cm}}{30 \text{ cm}} = \frac{4}{3} \]

So, the constant ratio of corresponding sides is \(\frac{4}{3}\).

\[ \frac{\text{AB}}{\text{DE}} = \frac{\text{BC}}{\text{EF}} = \frac{\text{CA}}{\text{FD}} = \frac{4}{3} \]

Determining the Ratio of Side Sums

We need to find the ratio of (BC + CA) to (EF + FD). From the side ratios we found:

\[ \frac{\text{BC}}{\text{EF}} = \frac{4}{3} \implies \text{BC} = \frac{4}{3} \times \text{EF} \]

\[ \frac{\text{CA}}{\text{FD}} = \frac{4}{3} \implies \text{CA} = \frac{4}{3} \times \text{FD} \]

Now, let's find the ratio of the sums:

\[ \frac{\text{BC} + \text{CA}}{\text{EF} + \text{FD}} = \frac{\left(\frac{4}{3} \times \text{EF}\right) + \left(\frac{4}{3} \times \text{FD}\right)}{\text{EF} + \text{FD}} \]

We can factor out \(\frac{4}{3}\) from the numerator:

\[ \frac{\frac{4}{3} (\text{EF} + \text{FD})}{\text{EF} + \text{FD}} \]

Assuming EF + FD is not zero (which it cannot be for a triangle), we can cancel out the term (EF + FD) from the numerator and the denominator:

\[ \frac{\text{BC} + \text{CA}}{\text{EF} + \text{FD}} = \frac{4}{3} \]

Thus, the ratio of (BC + CA) to (EF + FD) is 4:3.

Summary of the Solution Steps

1. Identify that the triangles are similar.
2. Understand that the ratio of perimeters of similar triangles equals the ratio of their corresponding sides.
3. Calculate the ratio of the given perimeters.
4. Apply this ratio to the corresponding sides.
5. Use the side ratios to find the ratio of the sum of specific sides.

The ratio of (BC + CA) to (EF + FD) is equal to the ratio of the perimeters of the two similar triangles.

Ratio of Perimeters = \(\frac{40}{30} = \frac{4}{3}\)

Since the triangles are similar, the ratio of corresponding sides is also \(\frac{4}{3}\).

The ratio of the sum of corresponding sides (like BC+CA and EF+FD) will also be the same as the ratio of individual corresponding sides and the ratio of perimeters.

Revision Table: Similar Triangles

Additional Information: Similar Triangles and Ratios

Similar triangles are triangles that have the same shape but potentially different sizes. Their corresponding angles are equal, and their corresponding sides are in proportion. This constant ratio of corresponding sides is often called the scale factor.

Understanding the relationships between side lengths, perimeters, and areas of similar triangles is fundamental in geometry. The fact that the perimeter ratio is linear (same as the side ratio) while the area ratio is quadratic (square of the side ratio) is a key concept.

In this specific problem, the question asks for the ratio of the sum of two sides (BC + CA) to the sum of the corresponding two sides (EF + FD). Since BC corresponds to EF and CA corresponds to FD, and their individual ratios are both equal to the scale factor (or perimeter ratio), their sum will also maintain that same ratio.

If \(\frac{\text{BC}}{\text{EF}} = k\) and \(\frac{\text{CA}}{\text{FD}} = k\), then \(\text{BC} = k \cdot \text{EF}\) and \(\text{CA} = k \cdot \text{FD}\).

So, \(\frac{\text{BC} + \text{CA}}{\text{EF} + \text{FD}} = \frac{k \cdot \text{EF} + k \cdot \text{FD}}{\text{EF} + \text{FD}} = \frac{k(\text{EF} + \text{FD})}{\text{EF} + \text{FD}} = k\).

The ratio is indeed equal to the ratio of corresponding sides, which we found to be the ratio of the perimeters, 4/3.