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Sagot :
[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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