23
Problem solving
Textbook: pp. 68–70
GoalSolve mixed problems on trapezoids, bisectors and proportional segments using similar triangles.
New words
intersection of the diagonals · diagonallar kesishish nuqtasisegment parallel to the bases · asoslarga parallel kesmaproportion · proporsiyasolution plan · masalani yechish rejasi
Explanation
In hard problems one finds a chain of pairs of similar triangles. An important result: in a trapezoid with bases a and b, the segment through the intersection of the diagonals parallel to the bases is bisected by that point and has length 2ab/(a + b). To get it, use the similarity of BOC and AOD to get OC : OA = a : b, then the similarity of ABC and AMO to write an equation for MO. Solution plan: figure, find and justify the similar pair, set up a proportion, find the unknown, check the answer.
Worked examples
In a trapezoid with bases a = 5 and b = 20, the parallel segment through the intersection of the diagonals is 2·5·20/(5 + 20) = 200/25 = 8.
DE ∥ AC, DB = 4, DE = 3, AC = 9: DB/AB = DE/AC = 1/3, so AB = 12.
Class activity
“Solution workshop”: groups solve one hard problem in 4 steps and write the reason for each step on a card; then they check another group's solution.
Practice
1
In a trapezoid with bases 4 and 12, what is the length of the segment through the intersection of the diagonals parallel to the bases?
6
2
DE ∥ AC, DB = 5, BA = 15 and DE = 4. How long is AC?
12
3
Two similar polygons have areas 5 dm² and 45 dm², and the sum of their perimeters is 48 dm. What is the larger perimeter (dm)?
36
4
Why do we first find the pair of similar triangles in a trapezoid problem?
The proportion between them links the unknown segment to the given data.