# Gradient, Midpoint And Distance

Before viewing this page, it would be helpful to learn Linear Graphs.

## Gradient or Slope

Gradient = | Rise |

Run |

where
**Rise** is the vertical height
**Run** is the horizontal height.

*Civil engineers use this formula to work out the gradient of roads. Carpenters use this formula to calculate the slope of a step.*

## Example One - Gradient

The world's most dangerous road is the Yungas Road in Bolivia in South America. More than 200 people are killed each year. Not only is it steep in parts but it is also in an area of high blinding rainfall. Calculate the gradient (steepness) of a part of the road with a rise of 215 metres and a run of 500 metres.

**Answer:**

Gradient = | Rise | = | 215 | = 0.43 |

Run | 500 |

## Gradient between Two Given Points

Gradient = m = | y_{2} – y_{1} |

x_{2} – x_{1} |

where (x_{1}, y_{1}) and (x_{2}, y_{2}) are two points on a linear or straight-line graph.

*Notice that the Rise is the difference between the y-values and the Run is the difference between the x-values.*

## Example Two - Positive Gradient

What is the gradient between the points (2,1) and (4,5)?

**Answer:**

(x_{1}, y_{1}) = (2,1)

(x_{2}, y_{2}) = (4,5)

Gradient = | y_{2} – y_{1} | = | 5 – 1 | = | 4 | = 2 |

x_{2} – x_{1} | 4 – 2 | 2 |

## Example Three - Negative Gradient

What is the gradient between the points (0,2) and (3,1)?

**Answer:**

(x_{1}, y_{1}) = (0,2)

(x_{2}, y_{2}) = (3,1)

Gradient = | y_{2} – y_{1} | = | 1 – 2 | = | –1 | = –^{1}⁄_{3} |

x_{2} – x_{1} | 3 – 0 | 3 |

*Notice that when the gradient is negative, the graph slopes backwards.*

## Questions - Find the Gradient between Two Points

**Q1.** (5,6) and (7,12)
**Q2.** (3,6) and (5,2)

**Answers**
**A1.** 3
**A2.** –2

## Gradients of Parallel Lines

Lines that are **parallel** have the **same gradient, m _{1} = m_{2}**

## Example Four - Parallel Lines

The graphs of y = 3x (red line) and y = 3x + 2 (blue line) are shown in the diagram. What do you notice about their gradients (slopes)?

**Answer:**

Same gradient

## Gradients of Perpendicular Lines

**Perpendicular** lines are **at right angles to each other**.

When the **gradients of perpendicular lines are multiplied**, the resulting product is **–1**.

**m _{1} × m_{2} = –1**

## Example Five - Perpendicular Lines

The graphs of y = 2x + 3 (blue line) and y = –^{1}⁄_{2} x + 5 (red line) are shown in the diagram. What do you notice about these graphs?

**Answer:**

Perpendicular (at right angles) to each other

## Example Six - Perpendicular Lines

If the gradient of a line is 2, what is the gradient of a line that is at right angles to it?

**Answer:**

–^{1}⁄_{2}

(2 multiplied by –^{1}⁄_{2} equals –1)

## Questions - Parallel or Perpendicular Lines

If the gradient of a line is 4, find the gradient of a line that is:
**Q1.** parallel
**Q2.** perpendicular.

**Answers**
**A1.** 4
**A2.** –^{1}⁄_{4}

## X-Intercept

The x-intercept is the point at which a graph crosses the x-axis.

The **y-coordinate at the x-intercept is 0**.

## Example Seven - X-Intercept

Find the x-intercept of the graph of y = x + 3.

**Answer:**

y = x – 3

0 = x – 3

x = 3

The coordinates of the x-intercept are (3,0)

## Y-Intercept

The y-intercept is the point at which a graph crosses the y-axis.

The **x-coordinate at the y-intercept is 0**.

## Example Eight - Y-Intercept

Find the y-intercept of the graph of y = x + 3.

**Answer:**

y = x – 3

y = 0 – 3

y = –3

The coordinates of the x-intercept are (0,–3)

## Questions - Intercepts

**Q1.** Find the x-intercept of y = 5x + 3.
**Q2.** Find the y-intercept of y = 4x – 8.

**Answers**
**A1.** 3
**A2.** 2

## Midpoint

The midpoint is the point that is **halfway** between 2 given points.

Midpoint = | (x_{1} + x_{2}) | , | (y_{1} + y_{2}) |

2 | 2 |

## Example Nine - Midpoint

What is the midpoint between the points (2,1) and (4,5)?

**Answer:**

(x_{1}, y_{1}) = (2,1)

(x_{2}, y_{2}) = (4,5)

Midpoint = | (x_{1} + x_{2}) | , | (y_{1} + y_{2}) |

2 | 2 |

Midpoint = | (2 + 4) | , | (1 + 5) |

2 | 2 |

Midpoint = (3,3)

## Questions - Find the Midpoint

**Q1.** (2,5) and (6, 10)
**Q2.** (–3, –5) and (–7, 9)

**Answers**
**A1.** (4 , 7.5)
**A2.** (–5, 2)

## Distance between Two Given Points

The distance between 2 points on a graph is really the **hypotenuse** of a right-angled triangle.

**Pythagoras's Rule** is used to find the distance between 2 points on a graph.

**Distance = √(x _{2} – x_{1}) ^{2} + (y_{2} – y_{1}) ^{2}**

## Example Ten - Distance between Points

What is the distance between the points (1,5) and (4,9)?

**Answer:**

(x_{1}, y_{1}) = (1,5)

(x_{2}, y_{2}) = (4,9)

Distance = √(x_{2} – x_{1}) ^{2} + (y_{2} – y_{1}) ^{2}

= √(4 – 1) ^{2} + (9 – 5) ^{2}

= √25

= 5

## Questions - Distance between Points

**Q1.** (2,5) and (7,17)
**Q2.** (2,1) and (4,5)

**Answers**
**A1.** 13
**A2.** 4.5