In the worksheet on division of linesegment student’s need to find the coordinates of the point dividing the line segment joining two given points in a given ratio.
Let us recall the formula for finding the coordinates 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 linesegment PQ internally in the ratio m : n, then the coordinates of R are {(mx₂ + nx₁)/(m + n) , (my₂ + ny₁)/(m + n)}.
(b) If the point R divides the linesegment PQ externally in the ratio m : n, then the coordinates of R are {(mx₂  nx₁)/(m  n), (my₂  ny₁)/(m  n)}.
To learn more about the formula for finding division of linesegment 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 coordinates of the point which divides the linesegment joining the points (2,  5) and ( 3,  2) externally in the ratio 4 : 3.
(iii) Find the coordinates of the point which divides the linesegment joining the, points ( x + y, x  y) and (x  y, x + y) internally in the ratio x : y.
(iv) Find the coordinates of the point which divides the linesegment 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 linesegment joining the points ( 3, 8) and (7,  7).
(ii) Find the ratio in which the point (5,  20) divides the linesegment 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 xaxis ? Also find the ratio in which it is divided by the yaxis.
4. (i) P is a point on the linesegment AB such that AP = 3 PB ; if the coordinates of A and B are (3, 4) and ( 5, 2) respectively, find the 1 coordinates of P.
(ii) The linesegment CD is produced to Q such that 2 CQ = 5 DQ; if the coordinates of C and D are (4, 7) and ( 2, 4) respectively, find the coordinates 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 coordinates (x, y) of Q. What are the coordinates of the midpoint of PQ?
5. If the point (0, 4) divides the linesegment joining the points ( 4, 10) and (2, 1) internally in a definite ratio, find the coordinate 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 coordinates of the terminal points.
7. Find the coordinates of the point of trisection of the linesegment joining the points ( 2, 3) and (3,  1) that is nearer to ( 2, 3).
8. Show that the linesegment joining the points (8, 3), ( 2, 7) and the linesegment 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 coordinates of the midpoints of the Indy, of a triangle, find the coordinates 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 linesegment 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)
● Coordinate Geometry
11 and 12 Grade Math
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