# HOW TO FIND THE VERTICES OF A TRIANGLE IF THE MIDPOINTS ARE GIVEN

Let D(x1, y1), E(x2, y2) and C(x3, y3) be the mid points of the sides AB, BC and CA of ΔABC.

Then, the vertices of ΔABC can be found as shown below.

A(x+ x- x2, y+ y- y2)

B(x+ x- x3, y+ y2 - y3)

C(x+ x- x1, y+ y3 - y1)

Example 1 :

The mid-points of the sides of a triangle are (5, 1), (3, -5) and (-5, -1). Find the coordinates of the vertices of the triangle.

Solution :

Let D, E and F be the mid points of the sides AB, BC and CA of ΔABC.

D(x1, y1) = (5, 1)

E(x2, y2) = (3, -5)

F(x3, y3= (-5, -1)

Vertex A :

A(x+ x- x2, y+ y- y2)

A(5 - 5 - 3, 1 - 1 - (-5))

A(-3, 5)

Vertex B :

B(x+ x- x3, y+ y2 - y3)

B(5 + 3 - (-5), 1 - 5 - (-1))

B(8 + 5, 1 - 5 + 1)

B(13, -3)

Vertex C :

C(x+ x- x1, y+ y3 - y1)

C(3 - 5 - 5, -5 - 1 - 1)

C(3 - 10, -5 - 2)

C(-7, -7)

Example 2 :

The mid-points of the sides of a triangle are (5, 3), (4, 0) and (2, 2). Find the coordinates of the vertices of the triangle.

Solution :

Let D, E and F be the mid points of the sides AB, BC and CA of ΔABC.

D(x1, y1) = (5, 3)

E(x2, y2) = (4, 0)

F(x3, y3= (2, 2)

Vertex A :

A(x+ x- x2, y+ y- y2)

A(5 + 2 - 4, 3 + 2 - 0)

A(3, 5)

Vertex B :

B(x+ x- x3, y+ y2 - y3)

B(5 + 4 - 2, 3 + 0 - 2)

B(7, 1)

Vertex C :

C(x+ x- x1, y+ y3 - y1)

C(4 + 2 - 5, 0 + 2 - 3)

C(1, -1)

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