Question

n the following​ figures, find the measure of the sides marked x. Note that the figures are not drawn to scale.
a.
4
4
A
B
D
x
C
E
7
Upper Delta​ABC~Upper DeltaADE
b.
A
B
C
x
3
D
E
F
7
9
Upper Delta​ABC~Upper DeltaEDF

Answer 1

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 \( 2.33 \))