7.1 Divisor of a Line Segment
A line segment is a portion of a straight line bounded by two specific endpoints with a measurable length. When an internal point lies on this line segment, it divides the segment into two distinct parts according to a specific ratio, \(m : n\). In coordinate geometry, understanding how this internal point partitions the segment allows us to derive a formula to compute the exact Cartesian coordinates of the dividing point, identify midpoints, and solve various real-world geometric problems involving motion and distance.
Learning Objectives
- Relate the position of an internal point dividing a line segment to the corresponding ratio \(m : n\).
- Derive and apply the divisor formula to determine the coordinates of a point dividing a line segment in a given ratio.
- Determine the coordinates of the midpoint of a line segment when \(m = n\).
- Calculate an unknown ratio or unknown coordinate of an endpoint given the dividing point.
- Solve real-life and geometric application problems involving the divisor of a line segment.
Geometric Division of a Line Segment on the Cartesian Plane
Visualise how horizontal and vertical projections form similar right-angled triangles to establish the divisor formula.
Projection on the \(x\)-axis
From similar triangles, \(\dfrac{x - x_1}{x_2 - x} = \dfrac{m}{n}\).
Rearranging gives \(x = \dfrac{nx_1 + mx_2}{m + n}\).
Projection on the \(y\)-axis
From similar triangles, \(\dfrac{y - y_1}{y_2 - y} = \dfrac{m}{n}\).
Rearranging gives \(y = \dfrac{ny_1 + my_2}{m + n}\).
Divisor of a line segment
\(P(x, y) = \left( \dfrac{nx_1 + mx_2}{m + n}, \dfrac{ny_1 + my_2}{m + n} \right)\)Cross-multiply: \(n\) pairs with \(A(x_1, y_1)\), \(m\) pairs with \(B(x_2, y_2)\).
Short Explanation
Understand the ratio, derive the divisor formula, and apply it to midpoints and practical problems.
1. Concept of a Divisor and Ratio
A line segment is part of a straight line with two endpoints. Any point that lies on and divides the segment into a particular ratio is an internal point.
If \(P\) divides \(AB\) in the ratio \(m : n\), then \(AP : PB = m : n\), and the total number of equal parts is \((m + n)\).
2. Derivation of the Divisor Formula
For \(A(x_1, y_1)\) and \(B(x_2, y_2)\), with \(P(x, y)\) dividing \(AB\) in the ratio \(m : n\):
\[x = \dfrac{nx_1 + mx_2}{m + n}, \quad y = \dfrac{ny_1 + my_2}{m + n}\]
3. Midpoint Formula
When \(m = n\), point \(P\) becomes the midpoint \(M\):
\[M = \left( \dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2} \right)\]
4. Determining Ratio or Unknown Coordinates
Substitute known coordinates into the \(x\)- or \(y\)-formula to form a linear relation in \(m\) and \(n\), then solve for the unknown ratio or endpoint coordinate.
5. Solving Practical Problems
For uniform velocity, if an object travels from \(A\) to \(B\) in total time \(T\) and reaches \(P\) in time \(t_1\), then \(AP : PB = t_1 : (T - t_1)\).
6. Trisection Points
Points that divide a line segment into three equal parts divide the line in the ratios \(1 : 2\) and \(2 : 1\).
| Quantity | Expression | Meaning |
|---|---|---|
| Distance \(AP\) | \(\dfrac{m}{m + n} AB\) | First part of the divided segment. |
| Distance \(PB\) | \(\dfrac{n}{m + n} AB\) | Second part of the divided segment. |
| Total equal parts | \(m + n\) | Number of equal parts across \(AB\). |
| Midpoint condition | \(m = n\) | Ratio \(1 : 1\), dividing the segment into two equal parts. |
Steps for Finding a Divisor Point
Identify \(A(x_1, y_1)\), \(B(x_2, y_2)\) and the ratio \(m : n\).
Apply the divisor formula for \(x\) and \(y\).
Simplify the numerator and denominator.
Write the coordinates of \(P(x, y)\).
Divisor formula
\(P(x, y) = \left( \dfrac{nx_1 + mx_2}{m + n}, \dfrac{ny_1 + my_2}{m + n} \right)\)Use this for any internal dividing point.
Midpoint formula
\(M = \left( \dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2} \right)\)Special case when \(m = n = 1\).
Try to Answer First
Answer in your mind, then press "Check Answer".
Find the coordinates of point \(P\) that divides the line segment joining \(A(-5, -2)\) and \(B(5, 8)\) in the ratio \(3 : 2\).
Point \(P(5, -3)\) divides the line segment \(AB\) in the ratio \(3 : 1\). If the coordinates of \(A\) are \((-7, 3)\), find the coordinates of point \(B(x, y)\).
Point \(P(-1, 2)\) lies on the straight line segment joining \(A(-2, 1)\) and \(B(2, 5)\). Determine the ratio \(AP : PB\).
Common Mistakes
- Swapping ratio terms: multiplying \(m\) with \(A(x_1, y_1)\) and \(n\) with \(B(x_2, y_2)\) instead of the correct cross-pairing: \(n(x_1, y_1)\) and \(m(x_2, y_2)\).
- Confusing fraction of total length with ratio of parts: treating "\(\frac{7}{9}\) of the distance from \(P\) to \(Q\)" as a ratio of \(7 : 9\) instead of \(PR : RQ = 7 : (9 - 7) = 7 : 2\).
- Sign calculation errors: dropping negative signs when multiplying negative coordinates, for example \(n(-x_1)\).
- Incorrect denominator: dividing by \((m - n)\) or \(2\) instead of the total number of parts, \((m + n)\).
Concept Misunderstandings
The midpoint formula is an entirely independent formula.
The midpoint formula is simply a special case of the divisor formula where \(m : n = 1 : 1\), resulting in \(\dfrac{1(x_1) + 1(x_2)}{1 + 1} = \dfrac{x_1 + x_2}{2}\).
The order of endpoints in ratio notation does not matter.
The ratio is directional. If \(P\) divides \(AB\) in the ratio \(m : n\), then \(AP : PB = m : n\). Reversing the endpoints to \(BA\) changes the division ratio to \(n : m\).
Summary
- An internal point \(P\) divides line segment \(AB\) into ratio \(AP : PB = m : n\), creating \((m + n)\) equal parts.
- Coordinates of divisor point \(P(x, y)\): \[P(x, y) = \left( \dfrac{nx_1 + mx_2}{m + n}, \dfrac{ny_1 + my_2}{m + n} \right)\]
- Midpoint coordinates when \(m = n = 1\): \[M = \left( \dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2} \right)\]
- Unknown ratios or endpoint coordinates are solved by separating the formula into independent linear equations for \(x\) and \(y\).
Short Activity
Objective: Apply the divisor formula, midpoint formula and ratio concepts.
A. Objective Quiz
1 Point \(P\) divides the line segment joining \(A(3, 7)\) and \(B(-7, 2)\) in the ratio \(3 : 2\). What are the coordinates of point \(P\)?
2 A ball is kicked from player \(P(4, 6)\) towards player \(Q(40, 45)\) and lands at \(\frac{2}{3}\) of the total distance along the straight line from \(P\) to \(Q\). In what ratio does the landing point divide the line segment \(PQ\)?
B. Fill in the Blanks
3 If a point \(P\) divides a line segment \(AB\) such that \(m = n\), then point \(P\) is the ________ of the line segment \(AB\).
4 In the formula for the divisor of a line segment \(P(x, y) = \left( \dfrac{nx_1 + mx_2}{D}, \dfrac{ny_1 + my_2}{D} \right)\), the denominator \(D\) represents the expression ________.
C. Matching / Drag and Drop
Drag each mathematical item to the correct description. On a phone, tap an item first, then tap the matching box.
