Question

Without using pythagoras theorem, show that the points (0,4) , (1,2) and (3,3) are the vertices of a right angle triangle.

Answer 1

Given points = ( 0, 4 ) ; ( 1 , 2 ) ; ( 3 , 3 )

$\text{Slope of any line = }\dfrac{y_{2} - y_{1}}{x_{2} - x_{1}}$

For the slope of ( 0 , 4 ) and ( 1 , 2 ),

For convenience, let's assume the points. On comparing the given situation with formula,

Taking x₁ = 0 , y₁ = 4 , x₂ = 1 , y₂ = 2

            Applying formula

⇒ Slope = $\dfrac{2-4}{1-0}$

⇒ Slope = - 2

For the slope of ( 1 , 2 ) and ( 3 , 3 ),

For convenience, let's assume the points. On comparing the given situation with formula,

Taking x₁ = 1 , y₁ = 2 , x₂ = 3 , y₂ = 3

            Applying formula

⇒ Slope = $\dfrac{3-2}{3-1}$

⇒ Slope = - 1 / 2

We know that two lines with negative of their reciprocal slopes are perpendicular to each other.

Hence, proved.

Answer 2

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We have, A0, 4; B1, 2 ; C3, 3 as the given points.We know that slope of straight line joining points x1, y1 and x2, y2 is ,slope = y2 - y1x2 - x1Now, slope of line AB,m1 = 2 - 41 - 0 = -2slope of line BC, m2 = 3 - 23 - 1 = 12slope of line CA, m3 = 4- 30- 3 =-13since, m1 × m2 = -2 × 12 = -1so, AB⊥BC.Hence , in ∆ABC, we have ∠B = 90°Hence, A0, 4; B1, 2 ; C3, 3 are the vertices of right ∆