I will solve each of the 8 problems on this worksheet using the Triangle Proportionality Theorem. This theorem states that if a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.
Let me go through each problem step by step.
Problem 1
Looking at the first diagram, we have a large right triangle with a smaller similar triangle inside it. The vertical side is divided into segments of length x and 12, and the horizontal base is divided into segments of length 6 and 8.
The proportion is: x/12 = 6/8
Solving for x:
x = 12 * (6/8)
x = 12 * 0.75
x = 9
Problem 2
In this diagram, we have a triangle with a line parallel to the base, creating two segments on the left side (3 and 15) and two segments on the right side (5 and x).
The proportion is: 3/15 = 5/x
Cross-multiplying:
3x = 15 * 5
3x = 75
x = 25
Problem 3
Here we have a triangle with a line parallel to the base. The left side has segments of 8 and 35, and the base has segments of x and 20.
The proportion is: 8/35 = x/20
Cross-multiplying:
35x = 8 * 20
35x = 160
x = 160/35
x = 32/7 ≈ 4.57
Problem 4
This diagram shows a triangle with a line parallel to the base. The left side has segments of 3 and 5, and the top side has segments of x and 9.
The proportion is: 3/5 = x/9
Cross-multiplying:
5x = 3 * 9
5x = 27
x = 27/5
x = 5.4
Problem 5
In this diagram, we have a triangle with a line parallel to the base. The left side has segments of 30 and 15, and the right side has segments of x and 20.
The proportion is: 30/15 = x/20
Simplifying:
2 = x/20
x = 40
Problem 6
This diagram shows a triangle with a line parallel to the base. The left side has segments of 6 and 8, and the right side has segments of 2 and x.
The proportion is: 6/8 = 2/x
Cross-multiplying:
6x = 8 * 2
6x = 16
x = 16/6
x = 8/3 ≈ 2.67
Problem 7
Here we have a triangle with a line parallel to the base. The left side has segments of 15 and 4, and the right side has segments of x and 10.
The proportion is: 15/4 = x/10
Cross-multiplying:
4x = 15 * 10
4x = 150
x = 150/4
x = 37.5
Problem 8
In this final diagram, we have a triangle with a line parallel to the base. The left side has segments of 4 and 5, and the right side has segments of 3 and x.
The proportion is: 4/5 = 3/x
Cross-multiplying:
4x = 5 * 3
4x = 15
x = 15/4
x = 3.75
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Final Answers:
1. x = 9
2. x = 25
3. x = 32/7 (or approximately 4.57)
4. x = 5.4
5. x = 40
6. x = 8/3 (or approximately 2.67)
7. x = 37.5
8. x = 3.75
Parent Tip: Review the logic above to help your child master the concept of triangle proportionality worksheet.
