Find the unknown side lengths in each pair of similar triangles. See Example 4.

Verified step by step guidance

1. Identify the pairs of corresponding sides in the similar triangles. Since the triangles are similar, their corresponding angles are equal, and their corresponding sides are proportional.
2. Set up a proportion between the lengths of the corresponding sides. For example, if the sides of the first triangle are \(a\), \(b\), and \(c\), and the corresponding sides of the second triangle are \(x\), \(y\), and \(z\), then the ratios \(\frac{a}{x} = \frac{b}{y} = \frac{c}{z}\) hold true.
3. Use the known side lengths to write an equation involving the unknown side length. For instance, if you know \(a\), \(b\), and \(x\), and want to find \(y\), use the proportion \(\frac{a}{x} = \frac{b}{y}\).
4. Solve the equation for the unknown side length by cross-multiplying and isolating the variable. This will give you an expression for the unknown side in terms of the known sides.
5. Repeat the process for any other unknown sides, ensuring that you always match corresponding sides and maintain the correct ratio from the similarity of the triangles.

Key Concepts

Similarity of Triangles

Two triangles are similar if their corresponding angles are equal and their corresponding sides are in proportion. This means the shape is the same but the size may differ, allowing us to set up ratios between corresponding sides to find unknown lengths.

Corresponding Sides and Angles

In similar triangles, each side corresponds to a side in the other triangle opposite the same angle. Identifying these pairs correctly is essential to apply proportional reasoning and solve for unknown side lengths accurately.

Setting Up and Solving Proportions

Once corresponding sides are identified, their lengths form ratios that are equal. Solving these proportions involves cross-multiplication and algebraic manipulation to find the unknown side lengths in one of the triangles.