Question

use the figure to complete the proportion.

1. \\(\frac{gc}{cf} = \frac{?}{db}\\)
2. \\(\frac{af}{fc} = \frac{?}{bd}\\)
3. \\(\frac{cd}{fb} = \frac{gd}{?}\\)
4. \\(\frac{ae}{cd} = \frac{ge}{?}\\)
5. \\(\frac{fg}{ag} = \frac{fb}{?}\\)
6. \\(\frac{gd}{ge} = \frac{?}{ae}\\)

use the given information to determine whether \\(\overline{bd} \parallel \overline{ae}\\).
7.
8.
9.
10.

determine the length of each segment.

1. \\(\overline{bc}\\)
2. \\(\overline{fc}\\)
3. \\(\overline{gb}\\)
4. \\(\overline{cd}\\)

Explanation:

Step1: Apply Triangle Proportionality Theorem (1)

For $\frac{GC}{CF}=\frac{?}{DB}$, use parallel lines $CD \parallel FB$, so $\frac{GC}{CF}=\frac{GD}{DB}$.

Step2: Apply Triangle Proportionality Theorem (2)

For $\frac{AF}{FC}=\frac{?}{BD}$, use parallel lines $FB \parallel AE$, so $\frac{AF}{FC}=\frac{BE}{BD}$.

Step3: Apply Triangle Proportionality Theorem (3)

For $\frac{CD}{FB}=\frac{GD}{?}$, use parallel lines $CD \parallel FB$, so $\frac{CD}{FB}=\frac{GD}{GB}$.

Step4: Apply Triangle Proportionality Theorem (4)

For $\frac{AE}{CD}=\frac{GE}{?}$, use parallel lines $CD \parallel AE$, so $\frac{AE}{CD}=\frac{GE}{GD}$.

Step5: Apply Triangle Proportionality Theorem (5)

For $\frac{FG}{AG}=\frac{FB}{?}$, use parallel lines $FB \parallel AE$, so $\frac{FG}{AG}=\frac{FB}{AE}$.

Step6: Apply Triangle Proportionality Theorem (6)

For $\frac{GD}{GE}=\frac{?}{AE}$, use parallel lines $CD \parallel AE$, so $\frac{GD}{GE}=\frac{CD}{AE}$.

Step7: Test parallelism (7)

Check $\frac{AB}{BC}=\frac{3}{6}=\frac{1}{2}$, $\frac{ED}{DC}=\frac{4}{8}=\frac{1}{2}$. Since ratios are equal, $\overline{BD} \parallel \overline{AE}$.

Step8: Test parallelism (8)

Check $\frac{AB}{BC}=\frac{1}{2}$, $\frac{ED}{DC}=\frac{1.2}{3}=0.4$. Ratios are not equal, so $\overline{BD}
parallel \overline{AE}$.

Step9: Test parallelism (9)

Check $\frac{AB}{BC}=\frac{5}{9}$, $\frac{ED}{DC}=\frac{3}{11}$. Ratios are not equal, so $\overline{BD}
parallel \overline{AE}$.

Step10: Test parallelism (10)

Check $\frac{AB}{BC}=\frac{5}{6}$, $\frac{ED}{DC}=\frac{7.5}{9}=\frac{5}{6}$. Since ratios are equal, $\overline{BD} \parallel \overline{AE}$.

Step11: Find $\overline{BC}$ length

Use $\frac{AB}{AE}=\frac{BC}{ED}$, $\frac{6}{5+5+7}=\frac{BC}{15}$, $\frac{6}{17}=\frac{BC}{15}$, so $BC=\frac{90}{17}\approx5.29$.

Step12: Find $\overline{FC}$ length

Use $\frac{AC}{AE}=\frac{FC}{ED}$, $\frac{6+6}{17}=\frac{FC}{15}$, $\frac{12}{17}=\frac{FC}{15}$, so $FC=\frac{180}{17}\approx10.59$.

Step13: Find $\overline{GB}$ length

Use $\frac{AG}{AE}=\frac{GB}{ED}$, $\frac{5}{17}=\frac{GB}{15}$, so $GB=\frac{75}{17}\approx4.41$.

Step14: Find $\overline{CD}$ length

$\overline{CD} = \overline{ED} - \overline{FC} = 15 - \frac{180}{17}=\frac{255-180}{17}=\frac{75}{17}\approx4.41$.

Answer:

1. $GD$
2. $BE$
3. $GB$
4. $GD$
5. $AE$
6. $CD$
7. $\overline{BD} \parallel \overline{AE}$
8. $\overline{BD}

parallel \overline{AE}$

1. $\overline{BD}

parallel \overline{AE}$

1. $\overline{BD} \parallel \overline{AE}$
2. $\frac{90}{17}$ (or $\approx5.29$)
3. $\frac{180}{17}$ (or $\approx10.59$)
4. $\frac{75}{17}$ (or $\approx4.41$)
5. $\frac{75}{17}$ (or $\approx4.41$)