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प्रश्न
A(3, 1), B(y, 4) and C(1, x) are vertices of a triangle ABC and G(3, 4) is its centroid. Find the values of x and y. Also, find the length of side BC.
उत्तर
Centroid of ΔABC = `((3 + y + 1)/3, (1+ 4 + x)/3) = ((4 + y)/3, (5 + x)/3)`
P, Q and R are the mid points of the sides BC, CA and AB.
By mid - point formula, we get
`=> P = ((y + 1)/2, (4 + x)/2), Q = (4/2, (1 + x)/2) and R = ((3 + y)/2, 5/2)`
Centroid of a ΔPQR = `(((y + 1)/2 + 4/2 + (3 + y)/2)/3, ((4 + x)/2 - (1 + x)/2 + 5/2)/3)`
`= (((y + 1 + 4 + 3 + y)/2)/3, ((4 + x + 1 + x + 5)/2)/3)`
`= ((8 + 2y)/6, (10 + 2x)/6)`
`= ((4 + y)/3, (5 + x)/3)` ........(ii)
From (i) and (ii), we get
Centroid of a ∆ABC = Centroid of a ∆PQR
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Find the coordinates of point P where P is the midpoint of a line segment AB with A(–4, 2) and B(6, 2).
Solution :
Suppose, (–4, 2) = (x1, y1) and (6, 2) = (x2, y2) and co-ordinates of P are (x, y).
∴ According to the midpoint theorem,
x = `(x_1 + x_2)/2 = (square + 6)/2 = square/2 = square`
y = `(y_1 + y_2)/2 = (2 + square)/2 = 4/2 = square`
∴ Co-ordinates of midpoint P are `square`.
