Problem Analysis:
The image shows two similar triangles, \( \triangle ABD \) and \( \triangle ABC \). The goal is to find the value of \( x \), which represents the height of the larger triangle \( \triangle ABC \).
Key Information from the Image:
1. \( \triangle ABD \) and \( \triangle ABC \) are similar triangles.
2. The height of \( \triangle ABD \) is given as 4 ft.
3. The base of \( \triangle ABD \) (segment \( BE \)) is 12 ft.
4. The base of \( \triangle ABC \) (segment \( BC \)) is the sum of \( BE \) and \( EC \), which is \( 12 \, \text{ft} + 6 \, \text{ft} = 18 \, \text{ft} \).
5. The height of \( \triangle ABC \) is \( x \).
Similar Triangles Property:
Since \( \triangle ABD \) and \( \triangle ABC \) are similar, the ratios of their corresponding sides are equal. Specifically:
\[
\frac{\text{Height of } \triangle ABD}{\text{Height of } \triangle ABC} = \frac{\text{Base of } \triangle ABD}{\text{Base of } \triangle ABC}
\]
Setting Up the Proportion:
Using the given information:
- Height of \( \triangle ABD \): 4 ft
- Base of \( \triangle ABD \): 12 ft
- Height of \( \triangle ABC \): \( x \)
- Base of \( \triangle ABC \): 18 ft
The proportion becomes:
\[
\frac{4}{x} = \frac{12}{18}
\]
Simplifying the Proportion:
Simplify the right-hand side of the equation:
\[
\frac{12}{18} = \frac{2}{3}
\]
So the proportion is:
\[
\frac{4}{x} = \frac{2}{3}
\]
Solving for \( x \):
Cross-multiply to solve for \( x \):
\[
4 \cdot 3 = 2 \cdot x
\]
\[
12 = 2x
\]
Divide both sides by 2:
\[
x = \frac{12}{2} = 6
\]
Final Answer:
The height of \( \triangle ABC \) is:
\[
\boxed{6}
\]
Parent Tip: Review the logic above to help your child master the concept of similar triangles.
