SOLUTION: Show that triangle ABC is right-angled using analytic geometry. Here are the coordinate points of the triangle. (0,0) and (-12,16) and (8.6). I have already managed to calculate it

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Question 1160710: Show that triangle ABC is right-angled using analytic geometry. Here are the coordinate points of the triangle. (0,0) and (-12,16) and (8.6). I have already managed to calculate it by solving the distance of each triangle, however, I recently learned that I must calculate the slope to answer this question and I am confused!!!
Found 3 solutions by jim_thompson5910, MathLover1, MathTherapy:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

For easier reference, label the three points
A = (0,0)
B = (-12,16)
C = (8,6)
It doesn't matter which letters you use, or what order you go with.

Let's find the slope of the line through points A and B
Use the slope formula
m = (y2-y1)/(x2-x1)
m = (16-0)/(-12-0)
m = 16/(-12)
m = -4/3
The slope of line AB is -4/3

Repeat for the slope of line BC
m = (y2-y1)/(x2-x1)
m = (6-16)/(8-(-12))
m = (6-16)/(8+12)
m = -10/20
m = -1/2
The slope of line BC is -1/2

Finally, compute the slope of line AC
m = (y2-y1)/(x2-x1)
m = (6-0)/(8-0)
m = 6/8
m = 3/4

----------------------------------------------
Recapping everything so far, we found these three slopes
slope of AB = -4/3
slope of BC = -1/2
slope of AC = 3/4

Now multiply each slope with one another
(slope AB)*(slope BC) = (-4/3)*(-1/2) = 2/3
(slope AB)*(slope AC) = (-4/3)*(3/4) = -12/12 = -1
(slope BC)*(slope AC) = (-1/2)*(3/4) = -3/8

The result in which we got -1 as a product is what we're after here. If two lines have their slopes multiply to -1, then those lines are perpendicular. This is assuming neither line is vertical.

The work above shows slopes AB and AC multiply to -1. They have the letter A in common. At the top of the page, I defined point A to be (0,0). This is where the 90 degree angle is located. Angle BAC, or CAB, is 90 degrees.

Diagram:

(diagram created with GeoGebra)

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
the coordinate points of the triangle.
let vertex A be at (0,0)
let vertex B be at (-12,16)
let vertex C be at (8,6)
distance AB represents side 1
distance AC represents side 2
distance BC represents side 3
find the length of the sides:
Solved by pluggable solver: Distance Between 2 points
The distance formula is sqrt%28%28%28x%5B2%5D-x%5B1%5D%29%5E2%29%2B%28%28y%5B2%5D-y%5B1%5D%29%5E2%29%29. Plug in the numbers,
sqrt%28%28%28-12-%280%29%29%5E2%29%2B%28%2816-%280%29%29%5E2%29%29
sqrt%28-12%5E2%2B16%5E2%29 The distance is 20.



=> side%5B1%5D=20

Solved by pluggable solver: Distance Between 2 points
The distance formula is sqrt%28%28%28x%5B2%5D-x%5B1%5D%29%5E2%29%2B%28%28y%5B2%5D-y%5B1%5D%29%5E2%29%29. Plug in the numbers,
sqrt%28%28%288-%280%29%29%5E2%29%2B%28%286-%280%29%29%5E2%29%29
sqrt%288%5E2%2B6%5E2%29 The distance is 10.



=> side%5B2%5D=10

Solved by pluggable solver: Distance Between 2 points
The distance formula is sqrt%28%28%28x%5B2%5D-x%5B1%5D%29%5E2%29%2B%28%28y%5B2%5D-y%5B1%5D%29%5E2%29%29. Plug in the numbers,
sqrt%28%28%288-%28-12%29%29%5E2%29%2B%28%286-%2816%29%29%5E2%29%29
sqrt%2820%5E2%2B-10%5E2%29 The distance is 22.3606797749979.



=> side%5B3%5D=22.3606797749979->keeping all decimal places for accuracy
to show that triangle ABC is right-angled, square of longest side must be equal to sum of squares other two sides
longest side is side%5B3%5D=22.3606797749979
%2822.3606797749979%29%5E2=20%5E2%2B10%5E2
500=500 which proves that triangle ABC is right-angled


Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
Show that triangle ABC is right-angled using analytic geometry. Here are the coordinate points of the triangle. (0,0) and (-12,16) and (8.6). I have already managed to calculate it by solving the distance of each triangle, however, I recently learned that I must calculate the slope to answer this question and I am confused!!!
You DON'T need to calculate the distance of each side to determine if it's a right-triangle. 

Calculating each slope and determining if any 2 are perpendicular to each other is enough to prove that the polygon is a right-triangle. And, perpendicularity, in this case, 

is proven if the PRODUCT of 2 of the slopes - 1, or one slope is the NEGATIVE RECIPROCAL of the other. 

In this case, one of the slopes is 3%2F4, and the other is -+4%2F3. As seen, EACH is the NEGATIVE RECIPROCAL of the other.