This worksheet is about the Triangle Proportionality Theorem, which states that if a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.
I will solve for `x` in each of the 8 problems by setting up and solving a proportion based on this theorem.
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Problem 1
- A line parallel to the base divides the left side into segments of length `x` and `12`, and the base into segments of `6` and `8`.
- The proportion is: $\frac{x}{12} = \frac{6}{8}$
- Solving for $x$: $x = 12 \times \frac{6}{8} = 12 \times \frac{3}{4} = 9$
Answer: x = 9
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Problem 2
- A line parallel to the right side divides the left side into segments of `3` and `15`, and the top side into segments of `5` and `x`.
- The proportion is: $\frac{3}{15} = \frac{5}{x}$
- Cross-multiplying: $3x = 75$
- Solving for $x$: $x = 25$
Answer: x = 25
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Problem 3
- A line parallel to the base divides the left side into segments of `8` and `x`, and the base into segments of `20` and `35`.
- The proportion is: $\frac{8}{x} = \frac{20}{35}$
- Simplify the right side: $\frac{20}{35} = \frac{4}{7}$
- So: $\frac{8}{x} = \frac{4}{7}$
- Cross-multiplying: $4x = 56$
- Solving for $x$: $x = 14$
Answer: x = 14
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Problem 4
- A line parallel to the right side divides the left side into segments of `3` and `5`, and the top side into segments of `x` and `9`.
- The proportion is: $\frac{3}{5} = \frac{x}{9}$
- Cross-multiplying: $5x = 27$
- Solving for $x$: $x = \frac{27}{5} = 5.4$
Answer: x = 5.4
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Problem 5
- A line parallel to the right side divides the left side into segments of `30` and `15`, and the bottom side into segments of `x` and `20`.
- The proportion is: $\frac{30}{15} = \frac{x}{20}$
- Simplify the left side: $\frac{30}{15} = 2$
- So: $2 = \frac{x}{20}$
- Solving for $x$: $x = 40$
Answer: x = 40
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Problem 6
- A line parallel to the right side divides the left side into segments of `6` and `8`, and the top side into segments of `2` and `x`.
- The proportion is: $\frac{6}{8} = \frac{2}{x}$
- Simplify the left side: $\frac{6}{8} = \frac{3}{4}$
- So: $\frac{3}{4} = \frac{2}{x}$
- Cross-multiplying: $3x = 8$
- Solving for $x$: $x = \frac{8}{3} \approx 2.67$
Answer: x = 8/3 or approximately 2.67
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Problem 7
- A line parallel to the right side divides the left side into segments of `15` and `4`, and the bottom side into segments of `x` and `10`.
- The proportion is: $\frac{15}{4} = \frac{x}{10}$
- Cross-multiplying: $4x = 150$
- Solving for $x$: $x = \frac{150}{4} = 37.5$
Answer: x = 37.5
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Problem 8
- A line parallel to the right side divides the left side into segments of `4` and `5`, and the top side into segments of `3` and `x`.
- The proportion is: $\frac{4}{5} = \frac{3}{x}$
- Cross-multiplying: $4x = 15$
- Solving for $x$: $x = \frac{15}{4} = 3.75$
Answer: x = 3.75
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Final Answers:
1.
x = 9
2.
x = 25
3.
x = 14
4.
x = 5.4
5.
x = 40
6.
x = 8/3 (≈ 2.67)
7.
x = 37.5
8.
x = 3.75
Parent Tip: Review the logic above to help your child master the concept of similar triangles word problems worksheet.
