In the worksheet on division of line-segment student’s need to find the co-ordinates of the point dividing the line segment joining two given points in a given ratio.
Let us recall the formula for finding the co-ordinates of the point dividing the line segment joining two given points in a given ratio as follows;
Let P (x₁, y₁) and Q (x₂, y₂) be two given points.
(a) If the point R divides the line-segment PQ internally in the ratio m : n, then the co-ordinates of R are {(mx₂ + nx₁)/(m + n) , (my₂ + ny₁)/(m + n)}.
(b) If the point R divides the line-segment PQ externally in the ratio m : n, then the co-ordinates of R are {(mx₂ - nx₁)/(m - n), (my₂ - ny₁)/(m - n)}.
To learn more about the formula for finding division of line-segment Click Here.
1. (i) If A and B be the points (1, 5) and (- 4, 7), then find the point P which divides AB internally in the ratio 2 : 3.
(ii) Find the co-ordinates of the point which divides the line-segment joining the points (2, - 5) and (- 3, - 2) externally in the ratio 4 : 3.
(iii) Find the co-ordinates of the point which divides the line-segment joining the, points ( x + y, x - y) and (x - y, x + y) internally in the ratio x : y.
(iv) Find the co-ordinates of the point which divides the line-segment joining the points (a, b) and (b, a) externally in the ratio (a - b) : (a + b).
2. (i) Find the ratio in which the point (1, 2) divides the line-segment joining the points (- 3, 8) and (7, - 7).
(ii) Find the ratio in which the point (5, - 20) divides the line-segment joining the points (4, 7) and (1, - 2).
3. In what ratio the segment joining the points (3, 4) and (2, - 3) is divided by the x-axis ? Also find the ratio in which it is divided by the y-axis.
4. (i) P is a point on the line-segment AB such that AP = 3 PB ; if the co-ordinates of A and B are (3, -4) and (- 5, 2) respectively, find the 1 co-ordinates of P.
(ii) The line-segment CD is produced to Q such that 2 CQ = 5 DQ; if the co-ordinates of C and D are (4, 7) and (- 2, 4) respectively, find the co-ordinates of Q.
(iii) If the point (6, 3) divides the segment of the line from P (4, 5) to Q (x, y) in the ratio 2 : 5, find the co-ordinates (x, y) of Q. What are the co-ordinates of the mid-point of PQ?
5. If the point (0, 4) divides the line-segment joining the points (- 4, 10) and (2, 1) internally in a definite ratio, find the co-ordinate of the point which divides the segment externally in the same ratio.
6. The straight line joining the points (2, - 2) and (4, 6) is extended each way a distance equal to half its own length. Determine the co-ordinates of the terminal points.
7. Find the co-ordinates of the point of trisection of the line-segment joining the points (- 2, 3) and (3, - 1) that is nearer to (- 2, 3).
8. Show that the line-segment joining the points (8, 3), (- 2, 7) and the line-segment joining (11, - 2), (5, 12) are bisected each other.
9. Find the lengths of the medians of the triangle whose vertices are (2, - 4), (6, 2) and (- 4, 2).
10. If (4, 3), (-2, 7) and ( 0, 11) are the co-ordinates of the mid-points of the Indy, of a triangle, find the co-ordinates of its vertices.
11. (i) Find (x, y) if (3, 2), (6, 3), (x, y) and (6, 5) are the vertices of a parallelogram taken in order.
(ii) If (x₁, y₁), (x₂, y₂), (x₃, y₃) and (x₄, y₄) be the consecutive vertices of dparallelogram, show that, x₁ + x₃ = x₂ + x₄ and y₁ + y₃ = y₂ + y₄.
Answers for the worksheet on division of line-segment are given below to check the exact answers of the above questions.
1. (i) (-1, 29/5)
(ii) (- 18, 7)
(iii)((x² + y²)/(x + y) ,(x² - y² + 2xy)/(x + y))
(iv) ((a² + b²)/2b, (b² - a² + 2ab)/2b).
2. (i) Internally in the ratio 2 : 3.
(ii) Externally in the ratio 3 : 2
3. Internally in the ratio 2 : 3. and externally in the ratio 3 : 2
4. (i) (-3, 1/2)
(ii) (-6 , 2)
(iii) Q (x, y) ≡ (11 – 2) , Mid – Point : (15/2, 3/2 )
5. (8, -8)
6. (5, 10) and (1, -6)
7. (-1/3 ,5/3)
9. √89, √17 and 5√2 units.
10. (6 , 7) , (2, -1) , (-6, 15)
11. (i) (x , y) = (9, 6)
● Co-ordinate Geometry
11 and 12 Grade Math
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