Coordinate Geometry and Points on SAT Math:Complete GuideWhat is Coordinate Geometry?Geometry always takes place on a plane, which is a flat surface that goes on infinitely in alldirections. The coordinate plane refers to a plane that has scales of measurement along the xand y-axes.

Coordinate geometry is the geometry that takes place in the coordinate plane.

Coordinate ScalesThe x-axis is the scale that measures horizontal distance along the coordinate plane.The y-axis is the scale that measures vertical distance along the coordinate plane.The intersection of the two planes is called the origin.

We can find any point along the infinite span of the plane by using its position with regard to the xand y-axes and to the origin. We mark this location with coordinates, written as (x, y).

The x value tells us how far along (and in which direction) our point is along the x-axis.

The y value tells us how far along (and in which direction) our point is along the y-axis.

For instance,

This point is 7 units to the right of the origin and 4 units above the origin. This means that our pointis located at coordinates (7, 4).

Anywhere to the right of the origin will have a positive x value. Anywhere left of the origin willhave a negative x value.Anywhere vertically above the origin will have a positive y value. Anywhere vertically below theorigin will have a negative y value.So, if we break up the coordinate plane into four quadrants, we can see that any point will havecertain properties in terms of its positivity or negativity, depending on where it is located.

Distances and MidpointsWhen given two coordinate points, you can find both the distance between them as well as themidpoint between the two original points. We can find these values by using formulas or by usingother geometry techniques.

Lets look at each option.

No distance is too much for a genius with a plan. Or a genius who is hungry. Either way.Image: Gwendal Uguen/Flickr

Distance Formula(x2x1)2+(y2y1)2

There are two options for finding the distance between two points--using the distance formula, orusing the Pythagorean Theorem. Lets look at both.

Solving Method 1--Distance FormulaIf you prefer to use formulas when you take standardized tests, then go ahead and memorize thedistance formula above. You will NOT be provided the distance formula on the test, so, if youchoose this route, make sure you can memorize the formula accurately and call upon it asneeded. (Remember--a formula you remember incorrectly is worse than not knowing a formulaat all!)

Let us say we have two points, (7, -2) and (-5, 3), and we must find the distance between the two.

If we simply plug our values into our distance formula, we get:

(x2x1)2+(y2y1)2

(57)2+(32)2

(12)2+(5)2

(144+25

169

13

The distance between our two points is 13.

Solving Method 2--Pythagorean Theorema2+b2=c2

Alternatively, we can always find the distance between two points by using the PythagoreanTheorem. This way takes slightly longer, but doesnt require us to expend energy memorizing extraformulas and carries less risk of us remembering the formula wrong.

Remember that you are given the Pythagorean Theorem on every SAT math section, so younever have to fear mis-remembering it. It is also a formula that youve likely had to use muchmore often than most other formulas, so odds are that its familiar to you.

Simply turn the coordinate points and the distance between them into a right triangle, with thedistance acting as a hypotenuse. From the coordinates, we can find the lengths of the legs of thetriangle and use the Pythagorean Theorem to find our distance.

For example, let us use the same coordinates from earlier to find the distance between themusing this method instead.

Find the distance between the points (7, -2) and (-5, 3)

First, start by mapping out your coordinates. Next, make the legs of your right triangles.

If we count the points along our plane, we can see that we have leg lengths of 12 and 5. Nowwe can plug these numbers in and use the Pythagorean Theorem to find the final piece of ourtriangle, the distance between our two points.

a2+b2=c2

122+52=c2

144+25=c2

169=c2

c=13

The distance between our two points is, once again, 13.[Special Note: If you are familiar with your triangle shortcuts, you may have noticed that thistriangle was what we call a 5-12-13 triangle. Because it is one of the regular right triangles, youtechnically dont even need the Pythagorean Theorem to know that the hypotenuse will be 13 ifthe two legs are 5 and 12. This is a shortcut that can be useful to know, but is NOT necessary toknow, as you can see.]

Midpoint Formula(

x1+x22,

y1+y22)

In addition to finding the distance between two points, we can also find the midpoint betweentwo coordinate points. Because this will be another point on the plane, it will have its own set ofcoordinates.

If you look at the formula, you can see that the midpoint is the average of each of the values ofa particular axis. So the midpoint will always be the average of the x values and the average ofthe y values, written as a coordinate point.

For example, let us take the same points we used for our distance formula, (7, -2) and (-5, 3).

If we take the average of our x values, we get:

7+52221

And if we take the average of our y values, we get:

2+3

120.5

The midpoint of the line will be at coordinates (1, 0.5).

If we look at our picture from earlier, we can see that this is true.

It is difficult to find the midpoint of a line without use of the formula, but by thinking of it as findingthe average of each axis value may make it easier to visualize and remember, rather thanthinking of it in terms of a formula.

Now, just measure themidpoint of an endless stretch of road--no problem.

Typical Point QuestionsPoint questions on the SAT will generally fall into one of three categories--questions about how thecoordinate plane works, counting questions, and midpoint or distance questions.

Lets look at each type.

Coordinate QuestionsQuestions about the coordinate plane test how well you understand exactly how the coordinateplane works, as well as how to manipulate points and lines within it.

For a question like this, it may be tempting to answer D, four. After all, there will be four distinctpoints 4 units from the origin, two on the x-axis (one right and one left), and two on the y-axis (oneup and one down).

But answering this way would disregard the realities of circles. Imagine that we have circle with amidpoint at the origin whose circumference touches each of the points 4 units from the origin.

Now, if we remember our circle definitions, we know that all straight lines drawn from the centerof the circle to the circumference will all be equal. We also know that there are infinite such lines.

This means that there will be infinitely many point that are 4 units from the origin. These points mayhave weird coordinates (as in non-integer values), but they will be points 4 units from the originall the same.

Our final answer is E, more than four.

Counting QuestionsCounting questions are exactly what they sound like--you will be given a diagram of thecoordinate plane (or, rarely, you must create your own) and then you will be asked to countdistances from specific point to specific point.

On occasion, you may also be asked to count seemingly "odd" measurements, like the values ofyour x and y coordinates.

For instance,

For this question, you must first understand what absolute values mean. From there, it is a simplematter of counting the x and y values from their coordinate points.

For a question like this, the most efficient path is to work from our answer choices. Since ouranswer choices are NOT in order of greatest to least, it will not help us to start with the middle

answer choice and work our way from there, as we would normally do when plugging in answers.Knowing that, let us simply work in order from first to last, until we find our right answer.

Point A is at coordinates (-3, -3). So let us find the sum of their absolute values.

|x|+|y|

|3|+|3|

3+3

6

Since we are looking for the value 5, this answer is too large. We can eliminate answer choice A.

Point B is at coordinates (-4, 1)

|x|+|y|

|4|+|1|

4+1

5

Success! We have found the answer choice that gives us coordinates whose absolute values addup to 5.

Because there will only ever be one correct answer on any SAT question, we can stop here.

Our final answer is B.

Midpoint and Distance QuestionsMidpoint and distance questions will be fairly straightforward and ask you for exactly that--thedistance or the midpoint between two points. You may have to find distances or midpoints froma scenario question (a hypothetical situation or a story) or simply from a straightforward mathquestion (e.g., What is the distance from points (4, 5) and (8, -2)?).

Lets look at an example of a scenario question,

Rosa and Marco met up for dinner and then drove home separately from the restaurant. To gethome from the restaurant, Rosa drove north 6 miles and Marco drove west 8 miles. How far apartdo Rosa and Marco live?

1.6 miles

2.8 miles

3.10 miles

4.12 miles

5.14 miles

First, let us make a quick sketch of our scenario.

Now, because this is a distance question, we have the option of using either our distance formulaor using the Pythagorean theorem. Since we have already begun by drawing out our diagram,let us continue on this path and use the Pythagorean theorem.

Now, we can see that we have made a right triangle from the legs of distance we have already.

Rosa drove 6 miles north and Marco drove 8 miles west, which means that the legs of our trianglewill be 6 and 8. Now we can find the hypotenuse by using the Pythagorean theorem.

62+82=c2

36+64=c2

100=c2c=100

c=10

[Note: if you remember your shortcuts for right triangles, you could have saved yourself some timeand simply known that our distance/hypotenuse was 10. Why? Because a right triangle with legsof 6 and 8 is a 3-4-5 triangle multiplied by 2. So the hypotenuse would be5*2=10.]

The distance between Marcos house and Rosas house is 10 miles.

Our final answer is C, 10 miles.

"The worst distance between two people is misunderstanding" - Unknown. Or, you know, 10 miles.

Strategies for Solving Point QuestionsThough point questions can come in a variety of forms, there are a few strategies you can followto help master them.

1) Always write down your given informationThough it may be tempting to work through questions in your head, it is easy to make mistakeswith your point questions if you do not write down your givens. This is especially the case whenworking with negatives or with absolute values.

In addition, most of the time you are given a diagram with marked points on the coordinateplane, you will not be given coordinates. This is because the test makers feel it would be toosimple a problem to solve had you been given coordinates (take, for example, the questioninvolving absolute values from earlier). So take a moment to write down your coordinates andany other given information in order to keep it straight in your head.

2) Draw it outIn addition to writing down your given information, draw pictures of your scenarios. Make yourown pictures if you are given none, draw on top of them if you are given diagrams. Neverunderestimate the value of marked information or a sketch--even a rough approximation canhelp you keep track of more information than you can (or should try to) in your head.

Time and energy are two precious recourses at your disposal when taking the SAT and it takeslittle of each to make a rough sketch, but can cost you both to keep all your information in yourhead.

3) Decide now whether or not to use formulasIf you feel more comfortable using formulas than using the slightly more drawn-out techniques,then decide now to memorize your formulas. Remember that memorizing a formula wrong isworse than not remembering it at all, so make sure that you memorize and practice your formulaknowledge between now and test day to lock it in your head.

If, however, you are someone who prefers to dedicate your study efforts elsewhere (or you simplyfeel that you wont remember the formula correctly on the day of the test), then go ahead andforget them. Use the Pythagorean theorem instead of memorizing the distance formula and washyour hands of memorization altogether.

There are multiple ways to solve most SAT math problems, so your choices should best match yourown personal strengths and weaknesses

Image: ljphillips34/Flickr

Test Your KnowledgeNow, lets test your point knowledge on some more real SAT math questions.1) What is the midpoint of the line that begins at coordinates (-3, 2) and ends at (5, -10)?

1.(6, -4)2.(4, -1)3.(1, 4)4.(-1, -6)5.(1, -4)

2)

3)

4)

Answers: E, D, A, BAnswer Explanations:1) To find the midpoint of the line connecting two points, we must take the average of each ofthe values along a particular axis.First, as always, it is a good idea to take a moment to map out the coordinates of our givenpoints.

This will help us keep track of our information, especially considering there are negatives involved.First, let us take the average of our two x-values.

3+52221Now, let us take the average of our two y values.

2+102824The midpoint of our line will be at coordinates (1, -4)We can see that this is likely the correct answer, as it neatly fits into our diagram.

Our final answer is E, (1, -4).

2) Here, we have a counting question. We are not being asked to find the linear distancebetween two points, F and W, but to find them along a grid. So let us draw the various pathwaysfrom F to W.

As you can see, the shortest paths from F to W are all 3

12units long, which makes 3

12the m-distance.Our final answer is D, 312.

3) Again, we have what amounts to another counting question. This is also a definite case ofwhen it is a good idea to draw pictures so that we do not repeat potential m-distance routesfrom F to Z.So let us find our routes. First, start by finding one of the most direct paths, which in this case is adistance of 4 units.

Next, trace all the paths that follow the lines from F to Z. If any of our new paths span lessthan 4units, it will of course become our new m-distance, but for now we are working under theassumption that the m-distance is 4.

All of our paths travel a distance of 4 units, making this our m-distance. If you were careful tokeep track of all your paths and not count any of them more than once, then you will see thatthere are 6 routes from F to Z that will measure the minimum distance.Our final answer is A, six.

4) Now, this question may seem tricky because it looks, at first glance, almost exactly like one ofour questions from earlier in the guide, which asked us, How many points are 4 units from theorigin?In that case, the answer was infinitely many, because all the points 4 units from the originformed a circle, and there are always infinite points on a circle.In this case, we are being asked to find all the points m-3-distance from a particular point. This isNOT the same as asking for the number of points 3 units from a point (in this case, point F). Whynot? Because the problem defined m-distance as the minimum distance traveled along a grid,not the distance in all directions.So if we start tracing all the distances m-3-units from F, we can start to see the pattern.

Once weve mapped out all the possible lines m-3-units from F in one quadrant of our map, wecan expand it outwards to see the shape that emerges.

We can see that all the points m-3-distance from F form a square.Our final answer is B, a square.

Think you deserve a treat for all that hard work.

The Take AwaysUnderstanding the coordinate plane and how points fit in it are the basic building blocks forcoordinate geometry. With these understandings, you will be able to perform more complexcoordinate geometry tasks, such as finding slopes and rotating shapes.

Coordinate geometry is not an insignificant part of the SAT math section, but luckily success ismostly a matter of organization and diligence. Be careful to keep track of your negatives and allyour moving pieces and youll be able to dominate those point questions and all the coordinategeometry the SAT can throw at you.