Question

How can you use the properties of similar triangles to solve for unknown lengths?

Answer 1

To solve for unknown lengths using the properties of similar triangles, you can follow these steps:

1. **Identify Similar Triangles**: Determine which triangles are similar. Triangles are similar if they have the same shape, which occurs when their corresponding angles are equal. This can be established through various criteria, such as Angle-Angle (AA), Side-Angle-Side (SAS), or Side-Side-Side (SSS) similarity.

2. **Set Up Corresponding Ratios**: Once you have identified the similar triangles, set up ratios using the lengths of their corresponding sides. If triangle \( ABC \) is similar to triangle \( DEF \), then the ratio of the lengths of corresponding sides is equal. For example:
\[
\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}
\]
Let’s say \( AB = a \), \( DE = d \), and you need to find side \( BC = b \) and corresponding side \( EF = e \).

3. **Write the Proportion**: If you have known lengths, you can write a proportion. For example, if \( AB = a \), \( DE = d \), and \( BC = b \) is the length you want to find, while \( EF = e \) is a corresponding length in triangle \( DEF \):
\[
\frac{AB}{DE} = \frac{BC}{EF} \Rightarrow \frac{a}{d} = \frac{b}{e}
\]

4. **Cross Multiply**: To solve for the unknown length, you can cross-multiply the proportion:
\[
a \cdot e = b \cdot d
\]
Rearranging this equation allows you to solve for \( b \):
\[
b = \frac{a \cdot e}{d}
\]

5. **Substitute Known Values**: Replace known values into the equation. This allows you to calculate the unknown length.

6. **Check Units**: Ensure that the units of measurement are consistent and appropriate for the context of the problem.

### Example
Suppose you have two similar triangles \( ABC \) and \( DEF \). If \( AB = 6 \), \( DE = 3 \), and you want to find \( BC \) when \( EF = 4 \):
1. Set up the proportion:
\[
\frac{6}{3} = \frac{BC}{4}
\]
2. Cross multiply:
\[
6 \cdot 4 = BC \cdot 3 \Rightarrow 24 = 3 \cdot BC
\]
3. Solve for \( BC \):
\[
BC = \frac{24}{3} = 8
\]

Thus, \( BC \) is 8 units long. Using the properties of similar triangles allows you to effectively solve for unknown lengths through proportional reasoning.