Coordinate Geometry and Points on SAT Math Complete Guide

Coordinate Geometry and Points on SAT Math Complete Guide SAT / ACT Prep Online Guides and Tips Coordinate geometry is one of the heavy-hitter topics on the SAT, and you'll need to be able to maneuver your way through its many facets in order to take on the variety of questions you'll see on the test. Luckily, though, coordinate geometry is not difficult to visualize or wrap your head around once you know the basics. And we are here to show you how. There will usually be two questions on any given SAT that involve points alone, and another 2-3 questions that will involve lines and slopes and/or rotations, reflections, or translations. This makes up a significant portion of your SAT math section, so it is a good idea to understand the ins and outs of coordinate geometry before you tackle the test. This will be your complete guide to points and the building blocks for coordinate geometry- how to find and manipulate points, distances, and midpoints, as well as strategies for solving these types of questions on test day. What is Coordinate Geometry? Geometry always takes place on a plane, which is a flat surface that goes on infinitely in all directions. The coordinate plane refers to a plane that has scales of measurement along the $x$- and $y$-axes. Coordinate geometry is the geometry that takes place in the coordinate plane. Coordinate Scales The $\bi x$-axis is the scale that measures horizontal distance along the coordinate plane. The $\bi y$-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 $x$- and $y$-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 point is located at coordinates $(7, 4)$. Anywhere to the right of the origin will have a positive $\bi x$ value. Anywhere left of the origin will have a negative $\bi x$ value. Anywhere vertically above the origin will have a positive $\bi y$ value. Anywhere vertically below the origin will have a negative $\bi y$ value. By breaking the coordinate plane up into four quadrants, we can see that any point will have certain properties in terms of its positivity or negativity, depending on where it is located. Distances and Midpoints When given two coordinate points, you can find both the distance between them as well as the midpoint between the two original points. We can find these values by using formulas or by using other geometry techniques. Let's 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 $√{(x_2−x_1)^2+(y_2−y_1)^2}$ There are two options for finding the distance between two points- using the distance formula, or using the Pythagorean Theorem. Let's look at both. Solving Method 1: Distance Formula If you prefer to use formulas when you take standardized tests, then go ahead and memorize the distance formula above. You will NOT be provided the distance formula on the test, so, if you choose this route, make sure you can memorize the formula accurately and call upon it as needed. (Remember- a formula you remember incorrectly is worse than not knowing a formula at 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: $√{(x_2−x_1)^2+(y_2−y_1)^2}$ $√{(−5−7)^2+(3−(−2))^2}$ $√{(−12)^2+(5)^2}$ $√{144+25}$ $√{169}$ $13$ The distance between our two points is 13. Solving Method 2: Pythagorean Theorem $a^2+b^2=c^2$ Alternatively, we can always find the distance between two points by using the Pythagorean Theorem. This way takes slightly longer, but doesn't require us to expend energy memorizing extra formulas and carries less risk of us remembering the formula wrong. Remember that you are given the Pythagorean Theorem on every SAT math section, so you never have to fear mis-remembering it. It is also a formula that you've likely had to use much more often than most other formulas, so odds are that it's familiar to you. Simply turn the coordinate points and the distance between them into a right triangle, with the distance acting as a hypotenuse. From the coordinates, we can find the lengths of the legs of the triangle and use the Pythagorean Theorem to find our distance. For example, let us use the same coordinates from earlier to find the distance between them using 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. Now we can plug these numbers in and use the Pythagorean Theorem to find the final piece of our triangle, the distance between our two points. $a^2+b^2=c^2$ $12^2+5^2=c^2$ $144+25=c^2$ $169=c^2$ $c=13$ 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 this triangle was what we call a 5-12-13 triangle. Because it is one of the regular right triangles, you technically don't even need the Pythagorean Theorem to know that the hypotenuse will be 13 if the two legs are 5 and 12. This is a shortcut that can be useful to know, but is NOT necessary to know, as you can see.] Midpoint Formula $$({x_1+x_2}/2, {y_1+y_2}/2)$$ In addition to finding the distance between two points, we can also find the midpoint between two coordinate points. Because this will be another point on the plane, it will have its own set of coordinates. If you look at the formula, you can see that the midpoint is the average of each of the values of a particular axis. So the midpoint will always be the average of the $x$ values and the average of the $y$ 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: $${7+(-5)}/2$$ $$2/2$$ $$1$$ And if we take the average of our $y$ values, we get: $${−2+3}/2$$ $$1/2$$ $$1/2$$ The midpoint of the line will be at coordinates $(1, 1/2)$. 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 finding the average of each axis value may make it easier to visualize and remember, rather than thinking of it in terms of a "formula." Now, just measure the midpoint of an endless stretch of road- no problem. Typical Point Questions Point questions on the SAT will generally fall into one of three categories- questions about how the coordinate plane works, counting questions, and midpoint or distance questions. Let's look at each type. Coordinate Questions Questions about the coordinate plane test how well you understand exactly how the coordinate plane works, as well as how to manipulate points and lines within it. In the $xy$-coordinate plane, how many points are a distance of 4 units from the origin? A. OneB. TwoC. FourD. More than four For a question like this, it may be tempting to answer C, four. After all, there will be four distinct points 4 units from the origin, two on the $x$-axis (one right and one left), and two on the $y$-axis (one up and one down). But answering this way would disregard the realities of circles. Imagine that we have circle with a midpoint 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 center of 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 may have "weird" coordinates (as in non-integer values), but they will be points 4 units from the origin all the same. Our final answer is D, More than 4. Counting Questions Counting questions are exactly what they sound like- you will be given a diagram of the coordinate plane (or, rarely, you must create your own) and then you will be asked to count distances from specific point to specific point. On occasion, you may also be asked to count seemingly "odd" measurements, like the values of your $x$ and $y$ coordinates. For instance, For this question, you must first understand what absolute values mean. From there, it is a simple matter 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 our answer 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 add up 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 Questions Midpoint and distance questions will be fairly straightforward and ask you for exactly that- the distance or the midpoint between two points. You may have to find distances or midpoints from a scenario question (a hypothetical situation or a story) or simply from a straightforward math question (e.g., "What is the distance from points $(4, 5)$ and $(8, -2)$?"). Let's look at an example of a scenario question, Rosa and Marco met up for dinner and then drove home separately from the restaurant. To get home from the restaurant, Rosa drove north 6 miles and Marco drove west 8 miles. How far apart do Rosa and Marco live? A. 8 milesB. 10 milesC. 12 milesD. 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 formula or 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 triangle will be 6 and 8. Now we can find the hypotenuse by using the Pythagorean theorem. $6^2+8^2=c^2$ $36+64=c^2$ $100=c^2$ c=√{100}$ $c=10$ [Note: if you remember your shortcuts for right triangles, you could have saved yourself some time and simply known that our distance/hypotenuse was 10. Why? Because a right triangle with legs of 6 and 8 is a 3-4-5 triangle multiplied by 2. So the hypotenuse would be $5*2=10$.] The distance between Marco's house and Rosa's house is 10 miles. Our final answer is B, 10 miles. "The worst distance between two people is misunderstanding"- Unknown. Or, you know, 10 miles. Strategies for Solving Point Questions Though point questions can come in a variety of forms, there are a few strategies you can follow to help master them. #1: Always Write Down Given Information Though it may be tempting to work through questions in your head, it is easy to make mistakes with your point questions if you do not write down your givens. This is especially the case when working with negatives or with absolute values. In addition, most of the time you are given a diagram with marked points on the coordinate plane, you will not be given coordinates. This is because the test makers feel it would be too simple a problem to solve had you been given coordinates (take, for example, the question involving absolute values from earlier). So take a moment to write down your coordinates and any other given information in order to keep it straight in your head. #2: Draw It Out In addition to writing down your given information, draw pictures of your scenarios. Make your own pictures if you are given none, draw on top of them if you are given diagrams. Never underestimate the value of marked information or a sketch- even a rough approximation can help you keep track of more information than you can (or should try to) in your head. Time and energy are two precious resourses at your disposal when taking the SAT and it takes little of each to make a rough sketch, but can cost you both to keep all your information in your head. #3: Decide Now Whether or Not to Use Formulas If 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 is worse than not remembering it at all, so make sure that you memorize and practice your formula knowledge 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 simply feel that you won't remember the formula correctly on the day of the test), then go ahead and forget them. Use the Pythagorean theorem instead of memorizing the distance formula and wash your hands of memorization altogether. There are multiple ways to solve most SAT math problems, so your choices should best match your own personal strengths and weaknesses Image: ljphillips34/Flickr Test Your Knowledge Now, let's 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 $(5, -10)$? A. (6, -4)B. (4, -1)C. (1, 4)D. (-1, -6)E. (1, -4) 2. 3. (Refer to information in question 2) 4. (Refer to information in question 2) Answers: E, D, A, B Answer Explanations: 1. To find the midpoint of the line connecting two points, we must take the average of each of the values along a particular axis. First, as always, it is a good idea to take a moment to map out the coordinates of our given points. 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. ${-3+5}/2$ $2/2$ 1 Now, let us take the average of our two $y$-values. ${2+(-10)}/2$ $-8/2$ $−4$ The 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 distance between two points, F and W, but to find them along a grid. So let us draw the various pathways from F to W. As you can see, the shortest paths from F to W are all 3 3$1/2$ units long, which makes 3$1/2$ the m-distance. Our final answer is D, 3$1/2$ 3. Again, we have what amounts to another counting question. This is also a definite case of when it is a good idea to draw pictures so that we do not repeat potential $m$-distance routes from F to Z. So let us find our routes. First, start by finding one of the most direct paths, which in this case is a distance of 4 units. Next, trace all the paths that follow the lines from F to Z. If any of our new paths span less than 4 units, it will of course become our new m-distance, but for now we are working under the assumption 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 to keep track of all your paths and not count any of them more than once, then you will see that there 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 of our questions from earlier in the guide, which asked us, "How many points are 4 units from the origin?" In that case, the answer was "infinitely many," because all the points 4 units from the origin formed 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 is NOT the same as asking for the number of points 3 units from a point (in this case, point F). Why not? 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 we've mapped out all the possible lines ${m-3}$-units from F in one quadrant of our map, we can 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 Aways Understanding the coordinate plane and how points fit in it are the basic building blocks for coordinate geometry. With these understandings, you will be able to perform more complex coordinate geometry tasks, such as finding slopes and rotating shapes. Coordinate geometry is not an insignificant part of the SAT math section, but luckily success is mostly a matter of organization and diligence. Be careful to keep track of your negatives and all your moving pieces and you'll be able to dominate those point questions and all the coordinate geometry the SAT can throw at you. What's Next? Ready to tackle more SAT math topics? You're in luck! We've got guides for every math topic on the SAT, so come check them out. From probabilities to polygons, fractions to functions- we've got you covered. Running out of time on the SAT math section? Check out our guide on how to beat the clock and maximize your SAT math score. Bitten by the procrastination bug? 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