Find the gradient of the line joining the points A(1, 2) and B(4, 11).

Write down the two points: A(1, 2), B(4, 11) Recall the gradient formula: m=(y_2-y_1)/(x_2-x_1) Substitute the coordinates: m=(11-(2))/(4-(1)) Simplify to find the gradient: m=3 Answer: 3

Find the gradient of the line joining the points A(-2, 5) and B(3, -5).

Write down the two points: A(-2, 5), B(3, -5) Recall the gradient formula: m=(y_2-y_1)/(x_2-x_1) Substitute the coordinates: m=(-5-(5))/(3-(-2)) Simplify to find the gradient: m=-2 Answer: -2

Find the gradient of the line joining the points A(-3, -1) and B(2, 4).

Write down the two points: A(-3, -1), B(2, 4) Recall the gradient formula: m=(y_2-y_1)/(x_2-x_1) Substitute the coordinates: m=(4-(-1))/(2-(-3)) Simplify to find the gradient: m=1 Answer: 1

Find the midpoint of the line segment joining A(2, 3) and B(8, 7).

Recall the midpoint formula: M=((x_1+x_2)/(2), (y_1+y_2)/(2)) Substitute the x-coordinates: (2+8)/(2)=5 Substitute the y-coordinates: (3+7)/(2)=5 State the midpoint: M=(5, 5) Answer: (5, 5)

Find the midpoint of the line segment joining A(-4, 1) and B(6, -3).

Recall the midpoint formula: M=((x_1+x_2)/(2), (y_1+y_2)/(2)) Substitute the x-coordinates: (-4+6)/(2)=1 Substitute the y-coordinates: (1+-3)/(2)=-1 State the midpoint: M=(1, -1) Answer: (1, -1)

Straight lines Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Straight lines questions. See exactly how to solve problems on gradient, coordinate geometry, midpoint, distance.

gradientcoordinate geometrymidpointdistancePythagorasequation of a line

A-Level70 questionsStep-by-step solutions

Question 1

3 markseasy

Find the gradient of the line joining the points

A\left(1,\ 2\right)

and

B\left(4,\ 11\right)

Worked solution

Write down the two points
$A\left(1,\ 2\right),\quad B\left(4,\ 11\right)$
Label the first point (x_1, y_1) and the second (x_2, y_2). Keeping them in order stops sign mistakes later.
Recall the gradient formula
$m=\frac{y_2-y_1}{x_2-x_1}$
The gradient measures steepness: how much the line goes up (or down) for each step to the right. This is the change in y divided by the change in x.
Substitute the coordinates
$m=\frac{11-(2)}{4-(1)}$
Carefully put each number in its place. Subtracting a negative becomes adding, so watch those double signs.
Simplify to find the gradient
$m=3$
Work out the top and bottom, then simplify the fraction. A positive answer slopes uphill, a negative one slopes downhill.

Answer

3

Question 2

3 markseasy

Find the gradient of the line joining the points

A\left(-2,\ 5\right)

and

B\left(3,\ -5\right)

Worked solution

Write down the two points
$A\left(-2,\ 5\right),\quad B\left(3,\ -5\right)$
Label the first point (x_1, y_1) and the second (x_2, y_2). Keeping them in order stops sign mistakes later.
Recall the gradient formula
$m=\frac{y_2-y_1}{x_2-x_1}$
The gradient measures steepness: how much the line goes up (or down) for each step to the right. This is the change in y divided by the change in x.
Substitute the coordinates
$m=\frac{-5-(5)}{3-(-2)}$
Carefully put each number in its place. Subtracting a negative becomes adding, so watch those double signs.
Simplify to find the gradient
$m=-2$
Work out the top and bottom, then simplify the fraction. A positive answer slopes uphill, a negative one slopes downhill.

Answer

-2

Question 3

3 markseasy

Find the gradient of the line joining the points

A\left(-3,\ -1\right)

and

B\left(2,\ 4\right)

Worked solution

Write down the two points
$A\left(-3,\ -1\right),\quad B\left(2,\ 4\right)$
Label the first point (x_1, y_1) and the second (x_2, y_2). Keeping them in order stops sign mistakes later.
Recall the gradient formula
$m=\frac{y_2-y_1}{x_2-x_1}$
The gradient measures steepness: how much the line goes up (or down) for each step to the right. This is the change in y divided by the change in x.
Substitute the coordinates
$m=\frac{4-(-1)}{2-(-3)}$
Carefully put each number in its place. Subtracting a negative becomes adding, so watch those double signs.
Simplify to find the gradient
$m=1$
Work out the top and bottom, then simplify the fraction. A positive answer slopes uphill, a negative one slopes downhill.

Answer

1

Question 4

2 markseasy

Find the midpoint of the line segment joining

A\left(2,\ 3\right)

and

B\left(8,\ 7\right)

Worked solution

Recall the midpoint formula
$M=\left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right)$
The midpoint is exactly halfway along the line segment. We simply average the x-values and average the y-values.
Substitute the x-coordinates
$\frac{2+8}{2}=5$
Add the two x-values together and split the result in half. This gives the x-coordinate of the middle point.
Substitute the y-coordinates
$\frac{3+7}{2}=5$
Do the same with the y-values. Adding then halving finds the height of the middle point.
State the midpoint
$M=\left(5,\ 5\right)$
Put the two answers together as a coordinate. That point sits dead centre between A and B.

Answer

\left(5,\ 5\right)

Question 5

2 markseasy

Find the midpoint of the line segment joining

A\left(-4,\ 1\right)

and

B\left(6,\ -3\right)

Worked solution

Recall the midpoint formula
$M=\left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right)$
The midpoint is exactly halfway along the line segment. We simply average the x-values and average the y-values.
Substitute the x-coordinates
$\frac{-4+6}{2}=1$
Add the two x-values together and split the result in half. This gives the x-coordinate of the middle point.
Substitute the y-coordinates
$\frac{1+-3}{2}=-1$
Do the same with the y-values. Adding then halving finds the height of the middle point.
State the midpoint
$M=\left(1,\ -1\right)$
Put the two answers together as a coordinate. That point sits dead centre between A and B.

Answer

\left(1,\ -1\right)

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