Ratio and Proportion Geometry Worksheet with problems on triangles, rectangles, and similar figures.

Here are the step-by-step solutions for each problem on the worksheet.

Problem 1

Given: Ratio $AC : AB = 3 : 4$. From the figure, side $AB = 16$ and side $AC = x$.

1. Set up the proportion based on the given ratio:
$$ \frac{AC}{AB} = \frac{3}{4} $$
2. Substitute the known values into the equation:
$$ \frac{x}{16} = \frac{3}{4} $$
3. Solve for $x$. To isolate $x$, multiply both sides by 16:
$$ x = \frac{3}{4} \times 16 $$
$$ x = 3 \times 4 $$
$$ x = 12 $$

Problem 2

Given: Ratio $AC : BC = 7 : 4$. From the figure, side $AC = 21$ and side $BC = 2x + 4$.

1. Set up the proportion:
$$ \frac{AC}{BC} = \frac{7}{4} $$
2. Substitute the expressions from the figure:
$$ \frac{21}{2x + 4} = \frac{7}{4} $$
3. Cross-multiply to solve for $x$:
$$ 21 \times 4 = 7 \times (2x + 4) $$
$$ 84 = 14x + 28 $$
4. Subtract 28 from both sides:
$$ 84 - 28 = 14x $$
$$ 56 = 14x $$
5. Divide by 14:
$$ x = \frac{56}{14} $$
$$ x = 4 $$

Problem 3

Task: Find the width to length ratio and simplify.
* Width = $10\text{ cm}$
* Length = $14\text{ cm}$

1. Write the ratio as a fraction:
$$ \frac{10}{14} $$
2. Simplify by dividing the top and bottom numbers by their greatest common divisor, which is 2:
$$ \frac{10 \div 2}{14 \div 2} = \frac{5}{7} $$
The ratio is 5 : 7.

Problem 4

Task: Find the width to length ratio and simplify.
* Width = $12\text{ m}$
* Length = $2\text{ m}$

*Note: Since the units are different (meters vs millimeters in other problems), we must ensure they match. Here both are in meters.*

1. Write the ratio:
$$ \frac{12}{2} $$
2. Simplify by dividing both by 2:
$$ \frac{12 \div 2}{2 \div 2} = \frac{6}{1} $$
The ratio is 6 : 1.

Problem 5

Task: Find the width to length ratio and simplify.
* Width = $16\text{ mm}$
* Length = $20\text{ mm}$

1. Write the ratio:
$$ \frac{16}{20} $$
2. Simplify by dividing both numbers by their greatest common divisor, which is 4:
$$ \frac{16 \div 4}{20 \div 4} = \frac{4}{5} $$
The ratio is 4 : 5.

Problem 6

Task: Find the scale factor of the smaller figure to the larger figure.
* Smaller triangle side = $14$
* Larger triangle side = $21$

1. The question asks for "smaller to larger," so we write the ratio as:
$$ \frac{\text{Smaller}}{\text{Larger}} = \frac{14}{21} $$
2. Simplify the fraction by dividing both numbers by 7:
$$ \frac{14 \div 7}{21 \div 7} = \frac{2}{3} $$
The scale factor is 2/3 (or 2:3).

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Final Answer:
① x = 12
② x = 4
③ 5 : 7
④ 6 : 1
⑤ 4 : 5
⑥ 2/3