Question

algebra solve for $x$.
1.

1. $x+4$
2. $13-x$

use the figure at the right to complete each proportion.

1. $\frac{a}{b} = \frac{\mathbf{\square}}{e}$
2. $\frac{b}{\mathbf{\square}} = \frac{c}{f}$
3. $\frac{f}{e} = \frac{c}{\mathbf{\square}}$
4. $\frac{a}{b+c} = \frac{\mathbf{\square}}{e+f}$

algebra solve for $x$.
8.
9.
10.

algebra solve for $x$.
11.
12.
13.
14.
15.
16.

Explanation:

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Problems 1-3 (Triangle Midsegment/Proportionality)

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Problem 1:

Step1: Set up proportionality

$\frac{10}{6} = \frac{x}{6}$

Step2: Solve for $x$

$x = 10$

Problem 2:

Step1: Set up proportionality

$\frac{2x}{x+4} = \frac{12}{9}$

Step2: Simplify ratio & solve

$\frac{2x}{x+4} = \frac{4}{3} \implies 6x = 4(x+4) \implies 6x=4x+16 \implies 2x=16 \implies x=8$

Problem 3:

Step1: Set up proportionality

$\frac{x}{13-x} = \frac{2}{3}$

Step2: Cross-multiply & solve

$3x = 2(13-x) \implies 3x=26-2x \implies 5x=26 \implies x=\frac{26}{5}=5.2$

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Problems 4-7 (Basic Proportions)

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Problem 4:

Step1: Match corresponding segments

$\frac{a}{b} = \frac{d}{e}$

Problem 5:

Step1: Match corresponding segments

$\frac{b}{c} = \frac{e}{f}$

Problem 6:

Step1: Match corresponding segments

$\frac{f}{e} = \frac{c}{b}$

Problem 7:

Step1: Apply addendo property

$\frac{a}{b+c} = \frac{d}{e+f}$

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Problems 8-10 (Parallel Line Proportionality)

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Problem 8:

Step1: Set up proportionality

$\frac{4}{6} = \frac{5}{x}$

Step2: Solve for $x$

$4x=30 \implies x=\frac{30}{4}=7.5$

Problem 9:

Step1: Set up proportionality

$\frac{4}{x} = \frac{6}{5}$

Step2: Solve for $x$

$6x=20 \implies x=\frac{20}{6}=\frac{10}{3}\approx3.33$

Problem 10:

Step1: Set up proportionality

$\frac{8}{12} = \frac{x}{24}$

Step2: Solve for $x$

$12x=192 \implies x=16$

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Problems 11-16 (Angle Bisector Theorem)

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Problem 11:

Step1: Apply Angle Bisector Theorem

$\frac{12}{x} = \frac{10}{5}$

Step2: Solve for $x$

$10x=60 \implies x=6$

Problem 12:

Step1: Apply Angle Bisector Theorem

$\frac{5}{8} = \frac{3}{x}$

Step2: Solve for $x$

$5x=24 \implies x=\frac{24}{5}=4.8$

Problem 13:

Step1: Apply Angle Bisector Theorem

$\frac{14}{x} = \frac{8}{20}$

Step2: Solve for $x$

$8x=280 \implies x=35$

Problem 14:

Step1: Apply Angle Bisector Theorem

$\frac{x}{6-x} = \frac{6}{4}$

Step2: Simplify & solve

$\frac{x}{6-x} = \frac{3}{2} \implies 2x=18-3x \implies 5x=18 \implies x=\frac{18}{5}=3.6$

Problem 15:

Step1: Apply Angle Bisector Theorem

$\frac{10-x}{x} = \frac{6}{8}$

Step2: Simplify & solve

$\frac{10-x}{x} = \frac{3}{4} \implies 40-4x=3x \implies 7x=40 \implies x=\frac{40}{7}\approx5.71$

Problem 16:

Step1: Apply Angle Bisector Theorem

$\frac{9}{x} = \frac{6}{8}$

Step2: Solve for $x$

$6x=72 \implies x=12$

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Answer:

1. $x=10$
2. $x=8$
3. $x=\frac{26}{5}=5.2$
4. $\frac{a}{b} = \frac{d}{e}$
5. $\frac{b}{c} = \frac{e}{f}$
6. $\frac{f}{e} = \frac{c}{b}$
7. $\frac{a}{b+c} = \frac{d}{e+f}$
8. $x=7.5$
9. $x=\frac{10}{3}\approx3.33$
10. $x=16$
11. $x=6$
12. $x=\frac{24}{5}=4.8$
13. $x=35$
14. $x=\frac{18}{5}=3.6$
15. $x=\frac{40}{7}\approx5.71$
16. $x=12$