Question

what is the value of x in the figure below? triangle figure with segments labeled 4, 2, 7, x and multiple - choice options: 3, 3.5, 4, 4.5

Explanation:

Step1: Identify Similar Triangles

The lines \( BE \) and \( CD \) are parallel (marked by arrows), so \( \triangle ABE \sim \triangle ACD \) by the Basic Proportionality Theorem (Thales' theorem).

Step2: Set Up Proportion

For similar triangles, the ratios of corresponding sides are equal. So, \( \frac{AB}{AC} = \frac{AE}{AD} \).
\( AB = 4 \), \( AC = AB + BC = 4 + 2 = 6 \), \( AE = 7 \), \( AD = AE + ED = 7 + x \).
Thus, \( \frac{4}{6} = \frac{7}{7 + x} \).

Step3: Solve for \( x \)

Cross - multiply: \( 4(7 + x)=6\times7 \)
\( 28 + 4x = 42 \)
Subtract 28 from both sides: \( 4x = 42 - 28 = 14 \)
Divide by 4: \( x=\frac{14}{4}=3.5 \)

Answer:

3.5