Question

in the figure below, assume the two lines appearing parallel are parallel. what is the length of \\(\overline{cg}\\)?

Explanation:

Step1: Identify Similar Triangles

Since \( BC \parallel FG \), triangles \( HBC \) and \( HFG \) are similar by the Basic Proportionality Theorem (Thales' theorem). For similar triangles, the ratios of corresponding sides are equal. Let \( CG = x \), then \( HG=HC + CG=20 + x \). The ratio of \( BC \) to \( FG \) should equal the ratio of \( HC \) to \( HG \). So, \(\frac{BC}{FG}=\frac{HC}{HG}\).

Step2: Substitute Values

We know \( BC = 14 \), \( FG = 21 \), \( HC = 20 \), and \( HG=20 + x \). Substituting these values into the proportion: \(\frac{14}{21}=\frac{20}{20 + x}\). Simplify \(\frac{14}{21}\) to \(\frac{2}{3}\), so we have \(\frac{2}{3}=\frac{20}{20 + x}\).

Step3: Cross - Multiply and Solve

Cross - multiplying gives \( 2(20 + x)=3\times20 \). Expand the left side: \( 40+2x = 60 \). Subtract 40 from both sides: \( 2x=60 - 40=20 \). Divide both sides by 2: \( x = 10 \).

Answer:

The length of \(\overline{CG}\) is 10.