# Coordinate Geometry

A new year symbolizes a new start for many.  Although the world is essentially the same as it was before the clock struck midnight, there is a new optimism about the future.  People want to focus on goals such as peace and prosperity.  Read this current event about an unexpected gesture from North Korea, and then ask yourself what you can expect from 2013.  What themes can you identify in this article that are likely to be part of an SAT essay question?

## Geometry: Coordinate Geometry

Read the following SAT test question and then select the correct answer.

Always read the question carefully and identify the bottom line.  Assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to verify that it addresses the bottom line.

In the figure, the slope of the line through points P and Q is $\frac{3}{2}$. What is the value of k?

Bottom Line: k = ?

Assess Your Options:  You could start from the point (1, 1) and use the slope to find new points, hoping that by adding 3 to the y value and 2 to the x value you will reach a point that contains a 7 y value.  Unfortunately, it is very easy to make a mistake using this method, such as adding the y change to the x value or vice versa.  Instead, use the information that you are given, the slope, to write an equation.

Attack the problem:  Although you are given the slope, you also know how the slope was obtained.  Think about it:  The slope is rise over run or the change in y over the change in x
$slope=\frac{rise}{run}=\frac{\bigtriangleup y}{\bigtriangleup x}=\frac{y_{2}\, -\, y_{1}}{x_{2}\, -\, x_{1}}$
You know two different y values, and two different x values, so you can plug in all the information that you know for the slope.
$slope =\frac{y_{2}\, -\, y_{1}}{x_{2}\, -\, x_{1}}=\frac{7-1}{k-1}$
Now you need to set this formula for slope equal to the value for slope that you were given in the problem, isolate the variable k, and solve for it.
$\frac{7-1}{k-1}=\frac{3}{2}$
$\frac{6}{k-1}=\frac{3}{2}$
3(k – 1) = 6 × 2
3k – 3 = 12
3k = 15
k = 5