"Do you remember this definition from the 'equations' unit?"

--TASK--

"Now stick the worksheet in your book and copy and complete the following observation table on the next page."

...midpoint between two coordinates from the equation.

Given the equation: y=2x+3

Randomly choose an x-value, say x=1, therefore, y=2(1)+3=5

So, Point A: (1,5)

Next, randomly choose a second x-value, say x=5, therefore, y=2(5)+3=13

So, Point B: (5,13)

Work out the midpoint: 

with two (x,y) coordinates: (1,5) and (5,13), we add the x values so 1 + 5 to get 6 then divide by 2 to get 3, next we add the y values so 5 + 13 to get 18 then divide by 2 to get 9, giving a midpoint coordinate of (3,9)

Below is the plotted line to show you.

...length of a line between two coordinates.

"Hang on, if we have two coordinates on a sloping line, that's like having the 'hypotenuse' on a right-angled triangle, right?" "So we can use Pythagoras's theory to work it out!"

So the theory states the following: Hypotenuse = √(Base² + Height²)

So, with two (x,y) coordinates: (1,5) and (5,13), we can work out the Base² as '4²' = '16' and the Height² as '8²' = '64' which when added together and rooted gives √80 = '8.94'

...equation from two coordinates.

In this type of question, we are given two coordinates and want to work out the equation firstly... 

STEP 1: Finding the 'm' value: The m-value of the equation is the slope, which we just saw is equal to the height/base, so when we have two (x,y) coordinates like (1,5) and (5,13), we use the same method as previously to work out the height as 8 and the base as 4 to get the slope of 2.

So we now have y = 2x + b

STEP 2: Finding the b-value: If one of the coordinates on the line is (1,5) (x = 1, when y = 5) and we know that the equation is in the form 2x + b, then we can enter these values to get 2(1) + b = 5 or 2 + b = 5, so b must = 3

Giving the equation of the line: y = 2x + 3

...equation from ONE coordinate + slope.

So you are given the m value and a coordinate (x1, y1)

To answer this, we need to use the 'POINT-SLOPE FORM.'

y - y1 = m(x - x1)

Next, substitute in the slope and the coordinate.

y - 3 = 2(x - 5) [Expand bracket]

y - 3 = 2x - 10 [+3]

y = 2x - 7

...equation of a parallel line from an equation and coordinate.

Parallel lines have the same slope, so if we are given the first equation:

y = 2x + 3, the slope is m = 2

If the new line passes through, say, (0,1)

Step 1: Use same slope

y = 2x + b

Step 2: Substitute the point

1 = 2(0) + b

Therefore, b = 1

Parallel equation:

y = 2x + 1

...equation of a perpendicular line from an equation & coordinate.

Perpendicular lines all have slopes that are 'negative reciprocals,' so if the original slope of the equation is m = 2, the perpendicular slope would be:

 m = −1/2

Now if the new line passes through (4,1)

Step 1: Use slope y = −1/2x + b

Step 2: Substitute the point's x and y-values of 4 and 1, then we get...

1 = −2 + b

Therefore, b = 3

Perpendicular equation:

y = −2x + 3

--TASK--

"Use the examples above to complete this worksheet!"

--TASK--

"Below are examples of spider diagrams based on two coordinates along a linear equation. You will get your own handout to complete, for display."