Advertisements
Advertisements
प्रश्न
Find the co-ordinates of centroid of a triangle if points D(–7, 6), E(8, 5) and F(2, –2) are the mid-points of the sides of that triangle.
उत्तर
Let A(x1, y1), B(x2, y2) and C(x3, y3) be the vertices of MBC.
Given that D(–7, 6), E(8, 5) and F(2, –2) are the midpoints of sides AB, BC and AC respectively.
Let G(x, y) be the centroid.
D is now the midpoint of AB.
∴ By midpoint formula.
–7 = `(x_1 + x_2)/2`
∴ x1 + x2 = –7 × 2
∴ x1 + x2 = –14 ......(i)
Similarly x2 + x3 = 16 ......(ii)
And x1 + x3 = 4 ......(iii)
Adding equations (i), (ii), and (iii), we get
x1 + x2 + x2 + x3 + x1 + x3 = –14 + 16 + 4
∴ 3x1 + 2x2 + 2x3 = 6
∴ 2(x1 + x2 + x3) = 6
∴ x1 + x2 + x3 = `6/2`
∴ x1 + x2 + x3 = 3 ......(iv)
D is the midpoint of AB.
∴ By the midpoint formula,
6 = `("y"_1 + "y"_2)/2`
∴ y1 + y2 = 12 ......(v)
Similarly, y2 + y3 = 10 ......(vi)
y1 + y3 = –4 ......(vii)
Adding equations (v), (vi), and (vii), we get
y1 + y2 + y2 + y3 + y1 + y4 = 12 + 10 – 4
∴ 2y1 + 2y2 + 2y3 = 18
∴ 2(y1 + y2 + y3) = 18
∴ y1 + y2 + y3 = `18/2`
∴ y1 + y2 + y3 = 9 ......(viii)
G(x, y) is the centroid of ΔABC.
∴ By centroid formula,
a = `(x_1 + x_2 + x_3)/3`
∴ x = `3/3` ......[From (iv)]
∴ x = 1
b = `("y"_1 + "y"_2 + "y"_3)/3`
∴ y = `9/3` ......[From (viii)]
∴ y = 3
APPEARS IN
संबंधित प्रश्न
ABCD is a parallelogram where A(x, y), B(5, 8), C(4, 7) and D(2, -4). Find
1) Coordinates of A
2) An equation of diagonal BD
In the given figure, P(4, 2) is mid-point of line segment AB. Find the co-ordinates of A and B.
Given a line ABCD in which AB = BC = CD, B = (0, 3) and C = (1, 8). Find the co-ordinates of A and D.
A(5, x), B(−4, 3) and C(y, –2) are the vertices of the triangle ABC whose centroid is the origin. Calculate the values of x and y.
In the following example find the co-ordinate of point A which divides segment PQ in the ratio a : b.
P(–3, 7), Q(1, –4), a : b = 2 : 1
Point P is the midpoint of seg AB. If co-ordinates of A and B are (-4, 2) and (6, 2) respectively then find the co-ordinates of point P.
(A) (-1,2) (B) (1,2) (C) (1,-2) (D) (-1,-2)
Find the midpoint of the line segment joining the following pair of point :
(4,7) and (10,15)
P( -2, 5), Q(3, 6 ), R( -4, 3) and S(-9, 2) are the vertices of a quadrilateral. Find the coordinates of the midpoints of the diagonals PR and QS. Give a special name to the quadrilateral.
(4, 2) and (-1, 5) are the adjacent vertices ofa parallelogram. (-3, 2) are the coordinates of the points of intersection of its diagonals. Find the coordinates of the other two vertices.
If (-3, 2), (1, -2) and (5, 6) are the midpoints of the sides of a triangle, find the coordinates of the vertices of the triangle.
A( 4, 2), B(-2, -6) and C(l, 1) are the vertices of triangle ABC. Find its centroid and the length of the median through C.
The coordinates of the centroid I of triangle PQR are (2, 5). If Q = (-6, 5) and R = (7, 8). Calculate the coordinates of vertex P.
The centre of a circle is (a+2, a-1). Find the value of a, given that the circle passes through the points (2, -2) and (8, -2).
P , Q and R are collinear points such that PQ = QR . IF the coordinates of P , Q and R are (-5 , x) , (y , 7) , (1 , -3) respectively, find the values of x and y.
A(3, 1), B(y, 4) and C(1, x) are vertices of a triangle ABC. P, Q and R are mid - points of sides BC, CA and AB respectively. Show that the centroid of ΔPQR is the same as the centroid ΔABC.
O(0, 0) is the centre of a circle whose one chord is AB, where the points A and B are (8, 6) and (10, 0) respectively. OD is the perpendicular from the centre to the chord AB. Find the coordinates of the mid-point of OD.
The mid-point of the line joining (−a, 2b) and (−3a, −4b) is
Find the coordinates of midpoint of segment joining (22, 20) and (0, 16)
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`.
