SOLUTION: Quadrilateral ABCD has vertices A(0,-2), B(9,1), C(4,6), and D(1,5). Prove: a) Quadrilateral ABCD is a trapezoid. b) Quadrilateral ABCD is an isosceles trapezoid.

Algebra ->  Algebra  -> Geometry-proofs -> SOLUTION: Quadrilateral ABCD has vertices A(0,-2), B(9,1), C(4,6), and D(1,5). Prove: a) Quadrilateral ABCD is a trapezoid. b) Quadrilateral ABCD is an isosceles trapezoid.      Log On

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Question 119374: Quadrilateral ABCD has vertices A(0,-2), B(9,1), C(4,6), and D(1,5). Prove: a) Quadrilateral ABCD is a trapezoid. b) Quadrilateral ABCD is an isosceles trapezoid.
Answer by Edwin McCravy(6936) About Me  (Show Source):
You can put this solution on YOUR website!
Quadrilateral ABCD has vertices A(0,-2), B(9,1), C(4,6), and D(1,5). Prove: a) Quadrilateral ABCD is a trapezoid. b) Quadrilateral ABCD is an isosceles trapezoid. 

drawing%28400%2C400%2C-5%2C10%2C-5%2C10%2C%0D%0A%0D%0Agraph%28400%2C400%2C-5%2C10%2C-5%2C10%29%2C%0D%0A%0D%0Aline%280%2C-2%2C9%2C1%29%2C+line%289%2C1%2C4%2C6%29%2C+line%284%2C6%2C1%2C5%29%2Cline%281%2C5%2C0%2C-2%29%2C%0D%0Alocate%28.3%2C-2%2CA%29%2C+locate%289%2C1%2CB%29%2C+locate%284%2C7%2CC%29%2C+locate%281%2C6%2CD%29+%29

First we prove that AB and DC are parallel, 
by showing they have the same slope:

Slope of AB
     
m = %28%281%29-%28-2%29%29%2F%28%289%29-%280%29%29 = %281%2B2%29%2F9 = 3%2F9 = 1%2F3

Slope of DC

m = %28%286%29-%285%29%29%2F%28%284%29-%281%29%29 = 1%2F3

There are both 1%2F3 so they are parallel and so ABCD is
a trapezoid.

To show that it is isosceles, we prove that AD = CB

Length of AD

d = sqrt%28%28%281%29-%280%29%29%5E2%2B%28%285%29-%28-2%29%29%5E2%29 = sqrt%28%281%29%5E2%2B%285%2B2%29%5E2%29 = sqrt%281%2B7%5E2%29 = sqrt%281%2B49%29
= sqrt%2850%29 = sqrt%2825%2A2%29 = 5sqrt%282%29 

Length of CB

Length of AD

d = sqrt%28%28%289%29-%284%29%29%5E2%2B%28%281%29-%286%29%29%5E2%29 = sqrt%28%285%29%5E2%2B%28-5%29%5E2%29 = sqrt%2825%2B25%29 = sqrt%2850%29 = sqrt%2825%2A2%29 = 5sqrt%282%29 

These are both 5sqrt%282%29 so it is also isosceles.

Edwin