Kuta Software Infinite Geometry Proportional Parts in Triangles and Parallel Lines Find the missing length indicated: Date Solve for Find the missing length indicated.
The Correct Answer and Explanation is:
To solve proportional parts in triangles involving parallel lines, we typically apply the Triangle Proportionality Theorem. This theorem states:
If a line is drawn parallel to one side of a triangle and intersects the other two sides, then it divides those two sides proportionally.
Problem Structure:
Suppose in triangle △ABC\triangle ABC△ABC, a line DEDEDE is drawn parallel to side BCBCBC, intersecting side ABABAB at point DDD and side ACACAC at point EEE. Then according to the theorem:ADDB=AEEC\frac{AD}{DB} = \frac{AE}{EC}DBAD=ECAE
Sample Problem:
Let’s say:
- AD=6AD = 6AD=6
- DB=3DB = 3DB=3
- AE=xAE = xAE=x
- EC=2EC = 2EC=2
We use the proportion:ADDB=AEEC\frac{AD}{DB} = \frac{AE}{EC}DBAD=ECAE
Substitute the known values:63=x2\frac{6}{3} = \frac{x}{2}36=2x
Simplify:2=x22 = \frac{x}{2}2=2x
Multiply both sides by 2:x=4x = 4x=4
Answer:
The missing length AE=4AE = 4AE=4 units.
Explanation
This problem involves using triangle similarity and proportional reasoning. When a line is drawn parallel to one side of a triangle and intersects the other two sides, it creates two smaller similar triangles within the larger one. These smaller triangles have proportional sides.
In our case, we apply the Triangle Proportionality Theorem, which tells us that the ratios of the corresponding segments on the intersected sides are equal. This principle is reliable whenever parallel lines divide sides of a triangle.
The problem gave us the lengths of three sides and asked us to find the fourth. First, we set up the proportion by comparing the parts on one side to the corresponding parts on the other. By isolating the unknown variable and solving the resulting simple equation, we found the missing length.
This method is consistent in all triangle problems with parallel lines intersecting two sides. It helps in geometry to break down complex figures into proportional relationships, making it easier to find missing dimensions.
In practical scenarios, such techniques are also used in map reading, architectural design, and engineering drawings where scale and proportion are crucial. Understanding proportional reasoning not only strengthens geometric insight but also builds a foundation for understanding more advanced topics in trigonometry and algebra.
By using ratios and logical steps, such problems can be solved efficiently without relying on complex formulas or memorization.
