Problem Analysis:
The given figure is a triangle \( \triangle ADE \) with a line segment \( BC \) parallel to the base \( DE \). This creates a smaller triangle \( \triangle ABC \) that is similar to \( \triangle ADE \). The problem asks us to find the length of the base \( DE \), denoted as \( x \).
Key Information from the Image:
- \( AB = 6 \, \text{cm} \)
- \( BC = 8 \, \text{cm} \)
- The height of \( \triangle ABC \) is 6 cm.
- The height of the trapezoid \( BCED \) is 9 cm.
Step-by-Step Solution:
####
Step 1: Use the properties of similar triangles.
Since \( BC \parallel DE \), \( \triangle ABC \) is similar to \( \triangle ADE \). This means the corresponding sides are proportional. Let the total height of \( \triangle ADE \) be \( h \). The height of \( \triangle ABC \) is 6 cm, and the height of the trapezoid \( BCED \) is 9 cm. Therefore, the total height \( h \) of \( \triangle ADE \) is:
\[
h = 6 + 9 = 15 \, \text{cm}
\]
####
Step 2: Set up the similarity ratio.
The ratio of the heights of the similar triangles \( \triangle ABC \) and \( \triangle ADE \) is:
\[
\frac{\text{Height of } \triangle ABC}{\text{Height of } \triangle ADE} = \frac{6}{15} = \frac{2}{5}
\]
Since the triangles are similar, the ratio of their corresponding bases is the same as the ratio of their heights. Therefore, the ratio of the base \( BC \) to the base \( DE \) is also \( \frac{2}{5} \):
\[
\frac{BC}{DE} = \frac{2}{5}
\]
Given that \( BC = 8 \, \text{cm} \), we can substitute this into the ratio:
\[
\frac{8}{x} = \frac{2}{5}
\]
####
Step 3: Solve for \( x \).
Cross-multiply to solve for \( x \):
\[
8 \cdot 5 = 2 \cdot x \implies 40 = 2x \implies x = \frac{40}{2} = 20
\]
Final Answer:
\[
\boxed{20}
\]
Parent Tip: Review the logic above to help your child master the concept of finding missing sides of similar triangles worksheet.
