SOLUTION: Show that the points A = (-2,0), B = (-4,4), and C = (8,5) are the vertices of a right triangle in two ways: (a)By using the converse of the Pythagorean Theorem (b)By usi

Algebra ->  Triangles -> SOLUTION: Show that the points A = (-2,0), B = (-4,4), and C = (8,5) are the vertices of a right triangle in two ways: (a)By using the converse of the Pythagorean Theorem (b)By usi      Log On


   

Question 1208013: Show that the points A = (-2,0), B = (-4,4), and C = (8,5) are the vertices of a right triangle in two ways:

(a)By using the converse of the Pythagorean Theorem
(b)By using the slopes of the lines joining the vertices
Found 2 solutions by josgarithmetic, math_tutor2020:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
You can handle the one using slopes yourself.

Using Pythagorean Theorem, apply distance formula to find these lengths:
AB=2sqrt%285%29;
AC=5sqrt%285%29;
BC=sqrt%28145%29;

and then expect %28sqrt%28145%29%29%5E2=%282sqrt%285%29%29%5E2%2B%285sqrt%285%29%29%5E2, which you will find is true.

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

The other tutor has discussed part (a).
I'll focus on part (b) only.

Use the slope formula to find the slope of line AB.




m+=+%28y%5B2%5D+-+y%5B1%5D%29%2F%28x%5B2%5D+-+x%5B1%5D%29

m+=+%284+-+0%29%2F%28-4+-+%28-2%29%29

m+=+%284+-+0%29%2F%28-4+%2B+2%29

m+=+%284%29%2F%28-2%29

m+=+-2
Line AB has slope -2.

Follow a similar process to find:
line BC has slope 1/12
line AC has slope 1/2

Compare slopes AB and AC.
They are -2 and 1/2 in that exact order.
The slopes multiply to -1, so it proves lines AB and AC are perpendicular.
They meet at a 90 degree angle.
This in turn proves triangle ABC is a right triangle (where the 90 degree angle is at point A).

See this page
https://www.algebra.com/algebra/homework/word/misc/Miscellaneous_Word_Problems.faq.question.1207909.html
for a proof as to why slopes that multiply to -1 lead to perpendicular lines.