3.1.4 Modelling with Straight Lines

Straight Lines

Gradient of a Line Segment

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The gradient $m$ of a line segment can be calculated using the formula:

m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}

where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.

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Example: Given points $(1, 2)$ and $(6, 10)$ :

Midpoint of a Line Segment

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The midpoint of a line segment is given by the mean of the $x$ and $y$ coordinates of its endpoints:

\left( \frac{1}{2} (x_1 + x_2), \frac{1}{2} (y_1 + y_2) \right)

where $(x_1, y_1)$ and $(x_2, y_2)$ are the endpoints.

Length of a Line Segment

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The length of a line segment can be found using the distance formula:
$\text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
where $(x_1, y_1)$ and $(x_2, y_2)$ are the endpoints.

Example Problem: Finding Coordinates of Points on a Line

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Problem: The line $l$ has gradient $-2$ and passes through the point $A(3, 5). \ \ B$ is a point on the line $l$ such that the distance $AB$ is $6\sqrt{5}$ . Find the coordinates of each of the possible points $B$ .

Solution:

Equation of Line $l$ :

The line has gradient $-2$ and passes through $(3, 5)$ . $y - 5 = -2(x - 3)$ $y - 5 = -2x + 6$ $y = -2x + 11$

Using Pythagoras' Theorem:

Let $B$ be $(x, -2x + 11)$ .
Distance $AB = 6\sqrt{5}$ : $\sqrt{(6 - 2x)^2 + (x - 3)^2} = 6\sqrt{5}$

Solve the Distance Equation: $(6 - 2x)^2 + (x - 3)^2 = 180$ $36 - 24x + 4x^2 + x^2 - 6x + 9 = 180$ $5x^2 - 30x + 45 = 180$ $5x^2 - 30x - 135 = 0$ $x^2 - 6x - 27 = 0$ $(x - 9)(x + 3) = 0$ $x = 9 \quad \text{or} \quad x = -3$
Find Corresponding $y$ -Coordinates:

For $x = 9$ : $y = -2(9) + 11 = -18 + 11 = -7$ $\therefore B(9, -7)$
For $x = -3$ : $y = -2(-3) + 11 = 6 + 11 = 17$ $\therefore B(-3, 17)$

Final Answers:

The coordinates of the possible points $B$ are $(9, -7)$ and $(-3, 17)$ .

Modelling with Straight Lines Simplified Revision Notes for A-Level OCR Maths Pure

3.1.4 Modelling with Straight Lines

Straight Lines

Gradient of a Line Segment

Midpoint of a Line Segment

Length of a Line Segment

Example Problem: Finding Coordinates of Points on a Line

Solution:

Final Answers:

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