# Parallel Lines Shortcut

Parallel lines in Analytical Geometry are two lines with the same slope. Basically, to solve for the equation of the line through a point and parallel to a given line with a known slope can be done in series of steps. However, using a shortcut formula, it can be done by one amazing move.

**Derivation:**

Given a line , find the equation of the line through .

The shortcut can be made if we observe a pattern when solving. Finding the line with a given conditions above has a pattern. Thus we can create a shortcut formula.

Step 1: Find the slope of the given line

From here, the slope of the given line is

Using the point-slope form, we can now solve for the equation of the required line.

But

** **

**Worked Problem 1:**

Find the equation of the line through and parallel to .

Solution:

Using the derived formula above we have,

**Worked Problem 2:**

Find the equation of the line through and parallel to the line .

Solution:

Using the derived formula we have,

**Worked Problem 3:**

is a rectangle with coordinates . Find the line containing the side .

Solution:

By drafting the figure as follows we have

We don’t have the coordinate of D but we can solve for the equation of the line

Solving for slope of ,

To solve for the equation of the line BC use either point B or C and use the point slope form.

We can now find the equation of the line AD using the shortcut formula above using point A and the line BC.

### Dan

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