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At National 5, there are a few skills required with straight lines, however, at Higher
these skills are expanded upon.
At National 5 there were three key formulae for straight lines, these were:
𝑦 2 −𝑦 1
• The Gradient Formula - 𝑥 2 − 𝑥 1
• The Equation of a Straight Line - 𝑦 = 𝑚𝑥 + 𝑐
• The Equation with Unknown Y-Intercept - 𝑦 − 𝑏 = 𝑚 (𝑥 − 𝑎)
At Higher, these formulae are the basis of complex problem-solving questions and
act as an introduction to new formulae.
Basic new formulae
At Higher, the first new formulae/ skills required are those of finding the distance
between two points on a straight line and finding the midpoint of a straight line.
These are both basic skills which are not assessed on their own but in the form of
larger more complex questions.
To find the distance, the Distance Formula is used, by subtracting the two x’s from
each other and the two y’s from each other, and then adding the two answers
together before square rooting that:
• √(𝑦 2 − 𝑦 1 ) + (𝑥 2 − 𝑥 1 )
, To find the midpoint of a straight line you add both x coordinates together and divide
them by 2 and do the same for the y coordinates. There is no formula for this skill as it
is only worth one mark.
Gradients & m=TANθ
At National 5 there were two gradient slopes, positive and negative, both of which gave
a numerical gradient and followed the equation 𝑦 = 𝑚𝑥 + 𝑐 ,however at Higher, there
are two other types of gradients a vertical and horizontal.
A vertical gradient runs parallel to the y axis and provides an undefined gradient (also
shown as 0 on denominator) as the slope is so minute, it is undefinable. Due to this a
vertical straight line does not follow the usual equation but instead follows: 𝑦 = 𝑥 .
A horizontal gradient runs parallel to the x axis and provides a zero gradient (also
shown as 0 on the numerator) as the line is running parallel to the axis and has no slope.
Due to this the line follows an equation of 𝑥 = 𝑦 .
The first higher formula for rectilinear shapes is, 𝑀 = 𝑡𝑎𝑛𝜃 this formula is used when
the gradient of a straight line creates an angle at the positive direction the x axis.
To apply 𝑀 = 𝑡𝑎𝑛𝜃 and calculate the angle at the positive direction of the x axis, there
are a series of steps that must be taken:
• Step 1: Use the gradient formula (from N5) to find the gradient
• Step 2: Substitute the gradient into the formula 𝑀 = 𝑡𝑎𝑛𝜃
• Step 3: Rearrange the formula to have the theta on the left and gradient on the
right.
• Step 4: Using inverse tan function find tan inverse of the gradient (this can be
non-calculator as well)
If the gradient is unknown but the angle is known the formula can be rearranged to
find 𝑀 .
parallel lines
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