Math Problem Statement
Solution
The problem describes a trapezoid , where , and provides several lengths. We need to find the length of segment .
Given Information:
We are asked to find the length of .
Steps:
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Use the Trapezoid Proportionality Theorem: Since , we can apply the trapezoid midsegment theorem or use the property that triangles formed by diagonals of the trapezoid are proportional.
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Identify Similar Triangles: Triangles and are similar by the AA similarity criterion because both share angle and the angles formed by the parallel lines and are congruent.
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Set Up Proportion: Since the triangles are similar, the sides are proportional. Thus, the ratio of corresponding sides is:
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Substitute Known Values:
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Solve for :
Thus, the length of is 11.25 units.
Would you like more details or clarifications? Here are five related questions you might find helpful:
- How can we use properties of trapezoids in other types of geometric problems?
- What is the trapezoid midsegment theorem, and when is it applicable?
- Can the similarity of triangles be applied in more complex shapes?
- How does proportional reasoning simplify calculations in geometry?
- What other geometric figures share properties similar to trapezoids?
Tip: Whenever working with parallel lines in polygons, check for similar triangles to simplify your calculations!
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Math Problem Analysis
Mathematical Concepts
Geometry
Similar Triangles
Proportions
Formulas
(SU / SR) = (TS / QR)
Theorems
Trapezoid Proportionality Theorem
AA Similarity Theorem
Suitable Grade Level
Grades 9-11