Question

1. find the value of \\(\overline{ar}\\) in the figure below if \\(\overline{re}\\) is parallel to \\(\overline{bc}\\).

Explanation:

Step1: Apply Basic Proportionality Theorem

Since \( \overline{RE} \parallel \overline{BC} \), by the Basic Proportionality Theorem (Thales' theorem), we have \( \frac{AR}{RB}=\frac{AE}{EC} \).
Substituting the given lengths: \( \frac{x - 4}{x - 1}=\frac{6}{12} \).

Step2: Simplify the Proportion

Simplify \( \frac{6}{12}=\frac{1}{2} \), so the equation becomes \( \frac{x - 4}{x - 1}=\frac{1}{2} \).
Cross - multiply: \( 2(x - 4)=1\times(x - 1) \).

Step3: Solve for \( x \)

Expand the left - hand side: \( 2x-8=x - 1 \).
Subtract \( x \) from both sides: \( 2x-x-8=x - x-1 \), which gives \( x - 8=-1 \).
Add 8 to both sides: \( x=-1 + 8=7 \).

Step4: Find the length of \( \overline{AR} \)

Since \( AR=x - 4 \) and \( x = 7 \), substitute \( x \) into the expression: \( AR=7-4 = 3 \).

Answer:

The value of \( \overline{AR} \) is \( 3 \).