Worksheet on Division of Line-Segment

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)}.

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

• What is Co-ordinate Geometry?
• Rectangular Cartesian Co-ordinates
• Polar Co-ordinates
• Relation between Cartesian and Polar Co-Ordinates
• Distance between Two given Points
• Distance between Two Points in Polar Co-ordinates
• Division of Line Segment: Internal & External
• Area of the Triangle Formed by Three co-ordinate Points
• Condition of Collinearity of Three Points
• Medians of a Triangle are Concurrent
• Apollonius' Theorem
• Problems on Distance Between Two Points
• Area of a Triangle Given 3 Points
• Worksheet on Rectangular – Polar Conversion
• Worksheet on Line-Segment Joining the Points
• Worksheet on Distance Between Two Points
• Worksheet on Distance Between the Polar Co-ordinates
• Worksheet on Finding Mid-Point
• Worksheet on Division of Line-Segment
• Worksheet on Centroid of a Triangle
• Worksheet on Area of Co-ordinate Triangle
• Worksheet on Collinear Triangle
• Worksheet on Area of Polygon
• Worksheet on Cartesian Triangle

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