See complete Problem 242 at:
gogeometry.com/problem/p242_triangle_equilateral_parallelogram.htm
Level: High School, SAT Prep, College geometry
Post your solutions or ideas in the comments.
Tuesday, February 3, 2009
Elearn Geometry Problem 242: Triangle with Equilateral triangles, Parallelogram
Labels:
congruence,
equilateral,
parallelogram,
triangle
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C'A = BA
ReplyDeleteangle C'AB' = Angle BAC = 60 - angle B'AB
AB' = AC
So, by SAS postulate,
triangle C'AB' is congruent to triangle BAC
So, C'B' = BC = BA'
Similarly as
triangle A'B'C is congruent to triangle BAC,
A'B' = BA = BC'
So,
in quadrilateral A'BC'B' both pairs of opposite sides are congruent.
So, it is a parallelogram.
http://img15.imageshack.us/img15/857/problem242a.png
ReplyDeleteSee the drawing
ReplyDelete- ΔACB’ and ΔBCA’ are equilateral => ∠ACB’ =∠BCA’ = Π/3
- ∠ACB = ∠ACB’ - ∠B’CB = Π/3 - ∠B’CB
- ∠B’CA’ = ∠BCA’ - ∠B’CB = Π/3 - ∠B’CB
- =>∠ACB =∠B’CA’
- ΔACB is congruent to ΔB’CA’ (SAS) => BA=B’A’
In the same way:
- ΔCAB’ and ΔBAC’ are equilateral => ∠CAB’ =∠BAC’ = Π/3
- ∠CAB = ∠CAB’ - ∠BAB’ = Π/3 - ∠BAB’
- ∠B’AC’ = ∠BAC’ - ∠BAB’ = Π/3 - ∠BAB’
- =>∠CAB =∠B’AC’
- ΔCAB is congruent to ΔB’AC’ (SAS) => BC=B’C’
- BA=B’A’ and BA=BC’ (ΔABC’ equilateral) => B’A’=BC’
- BC=B’C’ and BC=BA’ (ΔBCA’ equilateral) => B’C’=BA’
- => B’A’=BC’ and B’C’=BA’
- => B’A’BC’ is a parallelogram
Tr. ABC is congruent to AB'C' and also to A'B'C, SAS in each case.
ReplyDeleteSo B'C' = BC = A'B and also A'B' = AB = BC'
So the opposite sides of quadrilateral A'BC'B' are equal whence the same is a parallelogram
Sumith Peiris
Moratuwa
Sri Lanka