Question

prove thata(-2,2), b(3,7) anc c(6,4) form a right angled triangle. hence or otherwise, find it circumcentre and orthocentre

Answer 1

The circumcentre of a triangle is the point of intersection of the perpendicular bisectors of the sides of the triangle.

Let A(3,2),B(3,−2) and C(5,2) be the vertices of the triangle.

The midpoints of AB and AC are (3,0) and (4,2) respectively.

AB and BC are parallel to the Y axis and X axis respectively.

⇒ The perpendicular bisectors of AB and BC are parallel to the X axis and Y axis respectively.

⇒ The equations of the perpendicular bisectors of AB and BC are y=0 and x=4 respectively.

Solving the equations y=0 and x=4, we get the point of intersection of these two lines as (4,0).

⇒ The coordinates of the circumcentre of the triangle are (4,0).

Method 2

The circumcentre is the centre of the circumcircle of the triangle.

⇒ The circumcentre is equidistant from all the vertices of the triangle.

Let the coordinates of the circumcircle be (x,y).

The squares of the distances of the circumcentre from vertex A, vertex B and vertex C are (x−3)2+(y−2)2,(x−3)2+(y+2)2 and (x−5)2+(y−2)2 respectively.

⇒(x−3)2+(y−2)2=(x−3)2+(y+2)2=(x−5)2+(y−2)2.

⇒(y−2)2=(y+2)2 and (x−3)2=(x−5)2.

⇒−4y=4y and −6x+9=−10x+25.

⇒y=0 and x=4.

⇒ The coordinates of the circumcentre of the triangle are (4,0).

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Answer 2

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