What is the length of segment C'C'? What is the length of segment B'C'?

Steps to Solve

1. Identify Proportions from Similar Triangles

Since segments $B'C'$ and $BC$ are parallel, triangles $AC'C$ and $AB'C$ are similar. This means the ratios of their corresponding sides are equal.

2. Set Up Ratios

We know:

• The length of $C'A = 6$ units,
• The length of $AB' = 4$ units,
• The length of $AC = 10$ units,
• The length of $BC = 21$ units.

Thus, we can set up the following proportion to find $C'C'$:

$$ \frac{C'C'}{B'C} = \frac{C'A}{AB'} $$

Substituting the known lengths:

$$ \frac{C'C'}{4} = \frac{6}{10} $$

3. Cross-Multiply to Solve for (C'C')

Cross-multiplying gives us:

$$ 10 C'C' = 24 $$

Now, we solve for $C'C'$:

$$ C'C' = \frac{24}{10} = 2.4 \text{ units} $$

4. Find Length of Segment (B'C')

Next, we use the similar triangles again to find $B'C'$ using the ratio of the sides:

$$ \frac{B'C'}{BC} = \frac{AB'}{AC} $$

Substituting the known values:

$$ \frac{B'C'}{21} = \frac{4}{10} $$

5. Cross-Multiply to Solve for (B'C')

Cross-multiplying gives us:

$$ 10 B'C' = 84 $$

Now, we solve for $B'C'$:

$$ B'C' = \frac{84}{10} = 8.4 \text{ units} $$