Question

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\overline{bc} is parallel to \overline{de}.
what is ab?
enter your answer in the box.
\square units

Explanation:

Step1: Identify Similar Triangles

Since \( \overline{BC} \parallel \overline{DE} \), by the Basic Proportionality Theorem (Thales' theorem), \( \triangle ABC \sim \triangle ADE \). So, the ratios of corresponding sides are equal: \( \frac{AB}{AD}=\frac{AC}{AE} \).

Step2: Define Variables

Let \( AB = x \). Then \( AD = AB + BD = x + 17 \). \( AC = 42 \), \( AE = AC + CE = 42 + 21 = 63 \).

Step3: Set Up Proportion

Substitute into the proportion: \( \frac{x}{x + 17}=\frac{42}{63} \). Simplify \( \frac{42}{63}=\frac{2}{3} \), so \( \frac{x}{x + 17}=\frac{2}{3} \).

Step4: Solve for \( x \)

Cross - multiply: \( 3x = 2(x + 17) \). Expand: \( 3x = 2x + 34 \). Subtract \( 2x \) from both sides: \( 3x-2x=2x + 34-2x \), so \( x = 34 \).

Answer:

34