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Calculate the length of sides triangle pqr and determine weather or not triangle is a right angled. P(-4,6) q(6,1) r(2,9)

Answer :

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[tex]\bold{\huge{\underline{ Solution }}}[/tex]

Given :-

  • We have given the coordinates of the triangle PQR that is P(-4,6) , Q(6,1) and R(2,9)

To Find :-

  • We have to calculate the length of the sides of given triangle and also we have to determine whether it is right angled triangle or not

Let's Begin :-

Here, we have

  • Coordinates of P =( x1 = -4 , y1 = 6)
  • Coordinates of Q = ( x2 = 6 , y2 = 1 )
  • Coordinates of R = ( x3 = 2 , y3 = 9 )

By using distance formula

[tex]\pink{\bigstar}\boxed{\sf{Distance=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2\;}}}[/tex]

Subsitute the required values in the above formula :-

Length of side PQ

[tex]\sf{ = }{\sf\sqrt{ (6 - (-4))^{2} + (1 - 6)^{2}}}[/tex]

[tex]\sf{ = }{\sf\sqrt{ (6 + 4 )^{2} + (- 5)^{2}}}[/tex]

[tex]\sf{ = }{\sf\sqrt{ (10)^{2} + (- 5)^{2}}}[/tex]

[tex]\sf{ = }{\sf\sqrt{ 100 + 25 }}[/tex]

[tex]\sf{ = }{\sf\sqrt{ 125 }}[/tex]

[tex]\sf{ = 5 }{\sf\sqrt{ 5 }}[/tex]

Length of QR

[tex]\sf{ = }{\sf\sqrt{(2 - 6)^{2} + (9 - 1)^{2}}}[/tex]

[tex]\sf{ = }{\sf\sqrt{(- 4 )^{2} + (8)^{2}}}[/tex]

[tex]\sf{ = }{\sf\sqrt{16 + 64 }}[/tex]

[tex]\sf{ = }{\sf\sqrt{80 }}[/tex]

[tex]\sf{ = 4 }{\sf\sqrt{5 }}[/tex]

Length of RP

[tex]\sf{ = }{\sf\sqrt{ (-4 - 2 )^{2} + (6 - 9)^{2}}}[/tex]

[tex]\sf{ = }{\sf\sqrt{ (-6 )^{2} + (-3)^{2}}}[/tex]

[tex]\sf{ = }{\sf\sqrt{ 36 + 9 }}[/tex]

[tex]\sf{ = }{\sf\sqrt{ 45 }}[/tex]

[tex]\sf{ = 3}{\sf\sqrt{ 5 }}[/tex]

Now,

We have to determine whether the triangle PQR is right angled triangle

Therefore,

By using Pythagoras theorem :-

  • Pythagoras theorem states that the sum of squares of two sides that is sum of squares of 2 smaller sides of triangle is equal to the square of hypotenuse that is square of longest side of triangle

That is,

[tex]\bold{ PQ^{2} + QR^{2} = PR^{2}}[/tex]

Subsitute the required values,

[tex]\bold{ 125 + 80 = 45 }[/tex]

[tex]\bold{ 205 = 45 }[/tex]

From above we can conclude that,

  • The triangle PQR is not a right angled triangle because 205 ≠ 45 .

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