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

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)   (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)   (Show Source): You can put this solution on YOUR website!
the coordinate points of the triangle.
let vertex be at (,)
let vertex be at (,)
let vertex be at (,)
distance represents side
distance represents side
distance represents side
find the length of the sides:

Solved by pluggable solver: Distance Between 2 points

The distance formula is . Plug in the numbers,

The distance is 20.

=>

Solved by pluggable solver: Distance Between 2 points

The distance formula is . Plug in the numbers,

The distance is 10.

=>

Solved by pluggable solver: Distance Between 2 points

The distance formula is . Plug in the numbers,

The distance is 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

which proves that triangle is right-angled

Answer by MathTherapy(10552)   (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 , and the other is . As seen, EACH is the NEGATIVE RECIPROCAL of the other.
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