**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.

**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 = (y_{2} - y_{1})/(x_{2} - x_{1})

= (-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

CA^{2} = AB^{2} + BC^{2}

(√20)^{2 } = (√2)^{2} + (√18)^{2}

20 = 20

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

Solution :

Slope of LM = (y_{2} - y_{1})/(x_{2} - x_{1})

= (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 = √9^{2} + 7^{2}

= √130

MN = √6^{2} + 2^{2}

= √40

NL = √3^{2} + 9^{2}

= √90

LM^{2} = MN^{2} + NL^{2}

(√130)^{2 } = (√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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