Proportional Parts in Triangles and Parallel Lines worksheet with problems to find missing lengths and solve for x.

This image is a worksheet from "Kuta Software - Infinite Geometry" on the topic of "Proportional Parts in Triangles and Parallel Lines." The task is to find missing lengths or solve for variables using the properties of similar triangles and parallel lines.

The core principle used here is the Triangle Proportionality Theorem (also known as the Basic Proportionality Theorem or Thales' Theorem). It states that if a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those two sides proportionally.

In simpler terms, for a triangle with a line segment parallel to its base, the ratio of the segments on one side equals the ratio of the corresponding segments on the other side.

Let's solve each problem step-by-step.

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Problem 1

We have a triangle with a line parallel to the base. The left side is divided into segments of 15 and an unknown length (let's call it `x`). The base is divided into segments of 14 and 4.

According to the theorem:
`15 / x = 14 / 4`

Solving for `x`:
`15 * 4 = 14 * x`
`60 = 14x`
`x = 60 / 14 = 30 / 7 ≈ 4.29`

Answer: 30/7

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Problem 2

A line parallel to the base divides the left side into segments of 15 and 24, and the right side into segments of 25 and an unknown length (`x`).

Using the theorem:
`15 / 24 = 25 / x`

Solving for `x`:
`15 * x = 24 * 25`
`15x = 600`
`x = 600 / 15 = 40`

Answer: 40

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Problem 3

The left side is divided into 8 and 20, and the base is divided into 18 and an unknown length (`x`).

Using the theorem:
`8 / 20 = 18 / x`

Solving for `x`:
`8 * x = 20 * 18`
`8x = 360`
`x = 360 / 8 = 45`

Answer: 45

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Problem 4

The top side is divided into 7 and 3, and the right side is divided into 15 and 12.

Using the theorem:
`7 / 3 = 15 / 12`

Wait, this doesn't seem right. Let me re-examine the diagram. The line parallel to the base divides the left side into 7 and 3, and the right side into 15 and 12. So the segments are 7 and 3 on one side, and 15 and 12 on the other.

Actually, the correct proportion should be:
`7 / 3 = 15 / 12`

But `7/3 ≈ 2.33` and `15/12 = 1.25`, which are not equal. This suggests I might be misinterpreting the diagram. Let me assume the unknown is the total length of the base or something else. But the question asks for the missing length indicated, which seems to be the bottom part of the right side, which is given as 12. Perhaps the unknown is the total length of the right side? But 15 and 12 are both given.

Wait, perhaps the unknown is the bottom segment of the base? But the base is given as 15. I think there might be a mislabeling. Let me assume the unknown is the segment corresponding to 12, but it's already given. Perhaps the problem is to verify the proportion? But that doesn't make sense.

Alternatively, maybe the line parallel to the base divides the sides into 7 and 3 on the left, and 15 and 12 on the right, and we need to find the total length of the right side? But 15+12=27, which is given.

I think there might be a mistake in my interpretation. Let me move on and come back.

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Problem 5

We need to solve for `x`. The left side is divided into `5x` and 40, and the base is divided into 20 and 36.

Using the theorem:
`5x / 40 = 20 / 36`

Simplifying:
`5x / 40 = 5 / 9` (since 20/36 = 5/9)

So:
`5x = 40 * (5 / 9)`
`5x = 200 / 9`
`x = (200 / 9) / 5 = 200 / 45 = 40 / 9 ≈ 4.44`

Answer: 40/9

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Problem 6

We need to solve for `x`. The top side is divided into 28 and 8, and the right side is divided into `3x - 5` and 10.

Using the theorem:
`28 / 8 = (3x - 5) / 10`

Simplifying:
`28 / 8 = 7 / 2`

So:
`7 / 2 = (3x - 5) / 10`

Cross-multiplying:
`7 * 10 = 2 * (3x - 5)`
`70 = 6x - 10`
`80 = 6x`
`x = 80 / 6 = 40 / 3 ≈ 13.33`

Answer: 40/3

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Problem 7

We have a triangle with a line parallel to the base. The left side is divided into 15 and 5, and the right side is divided into 25 and an unknown length (`x`).

Using the theorem:
`15 / 5 = 25 / x`

Simplifying:
`3 = 25 / x`
`x = 25 / 3 ≈ 8.33`

Answer: 25/3

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Problem 8

We have a triangle with a line parallel to the base. The left side is divided into 77 and 7, and the right side is divided into 30 and 25.

Wait, this doesn't seem to fit the standard setup. Let me re-examine. The line parallel to the base divides the left side into 77 and 7, so the total left side is 84. The right side is divided into 30 and 25, so the total right side is 55. But the question asks for the missing length indicated, which seems to be the segment corresponding to 7 on the left, which is 25 on the right. But 25 is given.

Perhaps the unknown is the total length of the right side? But it's 30+25=55.

Alternatively, maybe the line parallel to the base divides the sides into 77 and 7 on the left, and 30 and an unknown on the right. But the diagram shows 30 and 25.

I think there might be a mislabeling. Let me assume the unknown is the segment corresponding to 7, which is given as 25. So perhaps the problem is to verify the proportion?

`77 / 7 = 11` and `30 / 25 = 1.2`, which are not equal. This suggests an error.

Alternatively, perhaps the segments are 77 and 7 on the left, and 30 and 25 on the right, and we need to find the total length of the base or something else. But the question asks for the missing length indicated, which is likely the 25, but it's given.

I think there might be a mistake in the diagram or my interpretation. Let me assume the unknown is the segment corresponding to 7, which is 25, so the answer is 25.

But that doesn't make sense. Perhaps the problem is to find the length of the base? But it's not indicated.

Given the ambiguity, I'll skip this for now.

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Summary of Answers:

1. 30/7
2. 40
3. 45
4. (Unclear, possibly 12 or 27)
5. 40/9
6. 40/3
7. 25/3
8. (Unclear, possibly 25)

For problems 4 and 8, the diagrams are ambiguous or potentially mislabeled. Based on standard interpretation, the answers for the clear problems are as above.