# HOW TO PROVE THE GIVEN VERTICES FORM A RIGHT TRIANGLE USING SLOPE

How to Prove the Given vertices form a Right Triangle Using Slope :

Here we are going to see, how to prove the given vertices form a right triangle.

In a right triangle one of the angles must be 90 degree. Let us look into some problems based on this concept.

## How to Prove the Given vertices form a Right Triangle Using Slope - Questions

Question 1 :

Show that the given vertices form a right angled triangle and check whether its satisfies pythagoras theorem

(i) A (1,-4) , B (2,-3) and C (4,-7)

Solution :

Slope of AB  =  (y2 - y1)/(x2 - x1)

=  (-3 - (-4))/(2 - 1)

=  (-3 + 4)/1

=  1

Slope of BC =  (-7 - (-3))/(4 - 2)

=  (-7 + 3)/2

=  4/2

=  2

Slope of CA =  (-7 - (-4))/(4 - 1)

=  (-7 + 4)/3

=  -3/3

=  -1

Slope of AB x Slope of CA  =  -1

1 (-1)  =  -1

-1  =  -1

Hence the given points are vertices of right triangle.

In order to check this with pythagoras theorem,  let us find the length of AB, BC and CA.

A (1,-4) , B (2,-3) and C (4,-7)

AB  =  √(-3+4)2 + (2-1)2

=  √2

BC  =  √(-7+3)2 + (4-2)2

=  √20

CA  =  √(-7+4)2 + (4-1)2

=  √18

CA2  =  AB2 + BC2

(√20) =  (√2)2 + (√18)2

20  =  20

(ii)  L (0, 5), M (9, 12) and N (3, 14)

Solution :

Slope of LM  =  (y2 - y1)/(x2 - x1)

=  (12 - 5)/(9 - 0)

=  7/9

Slope of MN =  (14 - 12)/(3 - 9)

=  2/(-6)

=  -1/3

Slope of NL =  (14 - 5)/(3 - 0)

=  9/3

=  3

Slope of MN x Slope of NL  =  -1

(-1/3) (3)  =  -1

-1  =  -1

Hence the given points are vertices of right triangle.

In order to check this with pythagoras theorem,  let us find the length of LM, MN and NL.

A (1,-4) , B (2,-3) and C (4,-7)

LM  =  √92 + 72

=  √130

MN  =  √62 + 22

=  √40

NL  =  √32 + 92

=  √90

LM2  =  MN2 + NL2

(√130) =  (√40)2 + (√90)2

130  =  130

Hence proved.

After having gone through the stuff given above, we hope that the students would have understood, "How to Prove the Given vertices form a Right Triangle Using Slope".

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