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प्रश्न
If the midpoints of the sides ofa triangle are (-2, 3), (4, -3), (4, 5), find its vertices.
उत्तर
Let P(x1, y1), Q(x2, y2 ) and R(x3, y3) be the coordinates of the vertices of .Δ PQR
Midpoint of PQ is D
D (-2 , 3) = D `(("x"_1 + "x"_2)/2 , ("y"_1 + "y"_2)/2)`
`("x"_1 + "x"_2)/2 = - 2 , ("y"_1 + "y"_2)/2 = 3`
X1 + X2 = -4 .....(1), Y1 + y2 = 6 .......(2)
similarly,
X2 + X3 = 8 ......(3), y2 + y3 = -6 ....(4)
X1+ X 3 = 8 ....(5), y1 +y3 = 10 ......(6)
Adding (1), (3) and (5)
2(x1 + x2 + x3 ) = 12
x1 + x2 + x3 = 6
- 4 + x3 = 6
x3 = 10
Adding (2), ( 4) and (6)
2(Y1+ Y2 + Y3) = 10
Y1+ Y2 + Y3 = 5
6 + Y3 = 5
Y3 = -1
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संबंधित प्रश्न
Calculate the co-ordinates of the centroid of the triangle ABC, if A = (7, –2), B = (0, 1) and C =(–1, 4).
The mid-point of the line segment joining (2a, 4) and (–2, 2b) is (1, 2a + 1). Find the values of a and b.
M is the mid-point of the line segment joining the points A(–3, 7) and B(9, –1). Find the coordinates of point M. Further, if R(2, 2) divides the line segment joining M and the origin in the ratio p : q, find the ratio p : q.
Write the co-ordinates of the point of intersection of graphs of
equations x = 2 and y = -3.
Three consecutive vertices of a parallelogram ABCD are A(S, 5), B(-7, -5) and C(-5, 5). Find the coordinates of the fourth vertex D.
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 .
Find the mid-point of the line segment joining the points
(a, b) and (a + 2b, 2a – b)
Find coordinates of the midpoint of a segment joining point A(–1, 1) and point B(5, –7)
Solution: Suppose A(x1, y1) and B(x2, y2)
x1 = –1, y1 = 1 and x2 = 5, y2 = –7
Using midpoint formula,
∴ Coordinates of midpoint of segment AB
= `((x_1 + x_2)/2, (y_1+ y_2)/2)`
= `(square/2, square/2)`
∴ Coordinates of the midpoint = `(4/2, square/2)`
∴ Coordinates of the midpoint = `(2, square)`
Find the coordinates of the mid-point of the line segment with points A(– 2, 4) and B(–6, –6) on both ends.
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`.
