To find the measure of the sides marked \( x \) in the two similar triangles provided, we will use the properties of similar triangles which state that corresponding sides are proportional.
a. \( \triangle ABC \sim \triangle ADE \)
We know:
- \( AB = 4 \)
- \( AC = 4 \)
- \( DE = 7 \)
- \( AE = x \)
Since the triangles are similar, we can set up a proportion based on the corresponding sides:
\[ \frac{AB}{AE} = \frac{AC}{DE} \]
Substituting the values we have:
\[ \frac{4}{x} = \frac{4}{7} \]
Now, cross-multiply to solve for \( x \):
\[ 4 \cdot 7 = 4 \cdot x \ 28 = 4x \ x = \frac{28}{4} = 7 \]
So, the value of \( x \) in part a is \( 7 \).
b. \( \triangle ABC \sim \triangle EDF \)
We know:
- \( AB = x \)
- \( AC = 7 \)
- \( DE = 3 \)
- \( EF = 9 \)
Setting up the proportion based on similar triangles:
\[ \frac{AB}{DE} = \frac{AC}{EF} \]
Substituting the known values:
\[ \frac{x}{3} = \frac{7}{9} \]
Cross-multiply:
\[ 9x = 21 \ x = \frac{21}{9} \ x = \frac{7}{3} \text{ or approximately } 2.33 \]
Thus, the value of \( x \) in part b is \( \frac{7}{3} \).
Summary:
- a. \( x = 7 \)
- b. \( x = \frac{7}{3} \) (approximately \( 2.33 \))