Advertisements
Advertisements
Question
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.
Solution
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
RELATED QUESTIONS
Points A and B have co-ordinates (3, 5) and (x, y) respectively. The mid-point of AB is (2, 3). Find the values of x and y.
(–5, 2), (3, −6) and (7, 4) are the vertices of a triangle. Find the length of its median through the vertex (3, −6).
Given a line ABCD in which AB = BC = CD, B = (0, 3) and C = (1, 8). Find the co-ordinates of A and D.
One end of the diameter of a circle is (–2, 5). Find the co-ordinates of the other end of it, if the centre of the circle is (2, –1).
P(4, 2) and Q(–1, 5) are the vertices of parallelogram PQRS and (–3, 2) are the co-ordinates of the point of intersection of its diagonals. Find co-ordinates of R and S.
Find the coordinates of midpoint of the segment joining the points (22, 20) and (0, 16).
Complete the table below the graph with the help of the following graph.
| Sr. No. | First point | Second point | Co-ordinates of first point (x1 , y1) | Co-ordinates of second point (x2 , y2) | `(y_2 - y_2)/(x_2 - x_2)` |
| 1 | C | E | (1, 0) | (3,4) | `4/2=2` |
| 2 | A | B | (-1,-4) | (0,-2) | `2/1 = 2` |
| 3 | B | D | (0,-2) | (2,2) | `4/2=2` |
Find th co-ordinates of the midpoint of the line segment joining P(0, 6) and Q(12, 20).
(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.
Find the centroid of a triangle whose vertices are (3, -5), (-7, 4) and ( 10, -2).
Two vertices of a triangle are (1, 4) and (3, 1). If the centroid of the triangle is the origin, find the third vertex.
A lies on the x - axis amd B lies on the y -axis . The midpoint of the line segment AB is (4 , -3). Find the coordinates of A and B .
The mid point of the line segment joining the points (p, 2) and (3, 6) is (2, q). Find the numerical values of a and b.
AB is a diameter of a circle with centre 0. If the ooordinates of A and 0 are ( 1, 4) and (3, 6 ). Find the ooordinates of B and the length of the diameter.
If P(–b, 9a – 2) divides the line segment joining the points A(–3, 3a + 1) and B(5, 8a) in the ratio 3: 1, find the values of a and b.
As shown in the figure. two concentric circles are given and line AB is the tangent to the smaller circle at T. Shown that T is the midpoint of Seg AB
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 coordinates of the point C dividing the line segment joining the points P(2, 4) and Q(5, 7) internally in the ratio 2 : 1 is
The coordinates of diameter AB of a circle are A(2, 7) and B(4, 5), then find the coordinates of the centre
