Question

# The mid points D,E,F of the sides of a triangle ABC are (3,4),(8,9)and(6,7) respectively. Find the vertices of the triangle.

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Solution

## Let A(x1,y1),B(x2,y2),C(x3,y3) be the vertices of △ABCWe have D is the midpoint of AB⇒(3,4)=(x1+x22,y1+y22)⇒x1+x2=6 ........(1) and y1+y2=8 ........(2)E is the midpoint of BC⇒(8,9)=(x2+x32,y2+y32)⇒x2+x3=16 ........(3) and y2+y3=18 ........(4)F is the midpoint of AC⇒(6,7)=(x1+x32,y1+y32)⇒x1+x3=12 ........(5) and y1+y3=14 ........(6)Equation (1)−(3) we getx1+x2−x2−x3=6−16⇒x1−x3=−10 .......(7)Equation (5)+(7) we getx1+x3+x1−x3=12−10=2⇒2x1=2 or x1=1Substituting the value of x1=1 in eqn(1) we getx1+x2=6 or x2=6−1=5Substituting the value of x1=1 in eqn(5) we getx1+x3=12 or x3=12−1=11Equation (2)−(4) we gety1+y2−y2−y3=8−18⇒y1−y3=−10 .......(8)Add equations (8) and (6) we gety1−y3+y1+y3=−10+14⇒y1=2From (2)y1+y2=8 or y2=8−2=6From (4)y1+y3=18 or y3=18−2=16∴ the co-ordinates of vertices of △ABC is A(1,2),(5,6),C(11,16)

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