Understanding Similar Triangles and Their Properties

The question asks us to find the length of side BC in triangle ABC, given that triangle ABC is similar to triangle EDF ($\Delta ABC \sim \Delta EDF$) and provided the lengths of sides AB, DF, and DE.

Similarity between two triangles is a fundamental concept in geometry. When two triangles are similar, it means they have the same shape but not necessarily the same size. This similarity implies two main properties:

• Their corresponding angles are equal.
• The ratio of their corresponding sides is constant.

Identifying Corresponding Sides in Similar Triangles

The notation $\Delta ABC \sim \Delta EDF$ is crucial for identifying the corresponding sides. The order of the vertices in the notation tells us which vertices correspond to each other:

• Vertex A corresponds to Vertex E
• Vertex B corresponds to Vertex D
• Vertex C corresponds to Vertex F

Based on these corresponding vertices, we can determine the corresponding sides:

• Side AB corresponds to side ED (or DE)
• Side BC corresponds to side DF
• Side AC corresponds to side EF

Setting up the Proportion using Side Ratios

Since the triangles are similar, the ratio of their corresponding sides is equal. We can write this as:

$$ \frac{AB}{ED} = \frac{BC}{DF} = \frac{AC}{EF} $$

We are given the following lengths:

• AB = 6 cm
• DF = 16 cm
• DE = 8 cm

We need to find the length of BC. Looking at the ratio equation, we can use the part that involves the known lengths and the unknown BC:

$$ \frac{AB}{ED} = \frac{BC}{DF} $$

Solving for the Length of BC

Now, let's substitute the given values into the proportion:

$$ \frac{6}{8} = \frac{BC}{16} $$

To solve for BC, we can cross-multiply or multiply both sides of the equation by 16:

$$ BC = \frac{6}{8} \times 16 $$

Simplify the fraction $\frac{6}{8}$ to $\frac{3}{4}$:

$$ BC = \frac{3}{4} \times 16 $$

Now, perform the multiplication:

$$ BC = 3 \times \frac{16}{4} $$

$$ BC = 3 \times 4 $$

$$ BC = 12 \text{ cm} $$

Thus, the length of BC is 12 cm.

Summary of Calculation for BC

Conclusion on BC Length

Based on the property of similar triangles that the ratio of corresponding sides is equal, and using the given lengths AB = 6 cm, DE = 8 cm, and DF = 16 cm, we calculated the length of BC to be 12 cm.

Revision Table: Similar Triangles

Additional Information: Scale Factor in Similar Triangles

The constant ratio between corresponding sides of similar triangles is called the scale factor. In this problem, the ratio $\frac{AB}{ED} = \frac{6}{8} = \frac{3}{4}$. This means that triangle ABC is $\frac{3}{4}$ times the size of triangle EDF in terms of side lengths. Conversely, triangle EDF is $\frac{ED}{AB} = \frac{8}{6} = \frac{4}{3}$ times the size of triangle ABC.

We found BC = 12 cm and DF = 16 cm. Let's check the ratio $\frac{BC}{DF} = \frac{12}{16} = \frac{3}{4}$. This confirms our calculation and the consistency of the scale factor.

The concept of scale factor is useful for finding any unknown side length if at least one pair of corresponding sides is known, providing the scale factor. It is also related to the ratio of areas (ratio of sides squared) and perimeters (same as ratio of sides).