To solve these problems, we’ll use the concept that similar triangles have proportional side lengths.

Problem (a)

The triangles are similar, so their corresponding sides are proportional. Set up a proportion between the given sides:

$\frac{82}{41} = \frac{x}{9}$

1. Simplify the left side: $\frac{82}{41} = 2$

2. Substitute this value into the equation: $2 = \frac{x}{9}$

3. Solve for $x$ by multiplying both sides by 9: $x = 2 \times 9 = 18$

So, the missing length $x = 18$.

Problem (b)

Here, we have two similar triangles, $\triangle ABC$ and $\triangle CDE$.

Given:

• $AB = x$
• $BC = 3$
• $CD = 4$
• $CE = 6$

Since these triangles are similar, we can set up the proportion: $\frac{AB}{CD} = \frac{BC}{CE}$

1. Substitute the known values: $\frac{x}{4} = \frac{3}{6}$

2. Simplify $\frac{3}{6}$ to $\frac{1}{2}$: $\frac{x}{4} = \frac{1}{2}$

3. Solve for $x$ by cross-multiplying: $x = \frac{1}{2} \times 4 = 2$

So, the missing length $x = 2$.

Would you like more details on these solutions?

Additional Questions

1. How would you solve if the similarity ratios were fractions?
2. What if one of the triangles had a different orientation?
3. How does knowing the triangles are similar help in finding unknown angles?
4. What if only one side length was known in each triangle?
5. How does proportionality work with non-right similar triangles?

Tip:

Always simplify ratios before solving proportions for easier calculations.