Question

If midpoints of the sides of ∆ABC are (1,2),(0,1)& (1,0), then find the coordinates of the vertices of ∆ABC.
(Class 10 Maths Sample Question Paper)

Answer 1

SOLUTION:
Let P(1,2), Q(0,1) & R(1,0) be the coordinates of the mid points of AB, BC,CA of  ∆ABC .Let the coordinates of the vertices of ∆ABC be A(x1,y1), B(x2,y2) and C(x3,y3).

P is the mid-point  of AB.

(x1+x2)/2= 1 and (y1+ y2)/2= 2

x1+x2 = 2 ………(1)
y1+y2= 4…………(2)

Q(0,1) is the mid-point of BC.

(x2+x3)/2= 0 and (y2+ y3)/2= 1

x2+x3 = 0………….(3)
y2+ y3= 2………….(4)

R(1,0) is the mid-point of AC.

(x3+x1)/2= 1 and (y3+ y1)/2= 0
x3+x1= 2…………(5)
y3+ y1= 0…………(6)

On adding eqs 1,3,& 5

2(x1+x2+x3)= 4
x1+x2+x3 = 4/2
x1+x2+x3 = 2……………(7)

On adding eqs 2, 4 and 6

2(y1+y2 +y3)= 6
y1+y2 +y3= 3…………….(8)

Subtracting eq 1 from eq 7

x1+x2+x3 = 2
x1+x2 = 2
(-)  (-)  (-)
---------------
x3 = 0

Subtracting eq 3 from eq 7

x1+x2+x3 = 2
x2+x3 = 0
(-)     (-)   (-)
---------------
x1 = 2

Subtracting eq 5 from eq 7

x1+x2+x3 = 2
x1    +x3 = 2
(-)   (-)    (-)
---------------
x2 = 0

Subtracting eq 2 from eq 8

y1+y2 +y3= 3
y1+y2       = 4
(-)   (-)    (-)
--------------------
y3            = - 1

Subtracting eq 4 from eq 8

y1+y2 +y3= 3
  y2+ y3= 2
  (-)    (-)   (-)
--------------------
y1      =  1

Subtracting eq 6 from eq 8

y1+y2 +y3= 3
y1     + y3= 0
(-)    (-)       (-)
--------------------
  y2       =  3

The value of x1 = 2, x2 = 0 and x3=0
The value of y1 = 1, y2 = 3 and y3= -1

Hence, the coordinates of the vertices of the ∆ABC are A(2,1), B(0,3), C(0,-1).

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