Coordinate Geometry on ACT Math Strategies and Practice

Coordinate Geometry on ACT Math Strategies and Practice SAT / ACT Prep Online Guides and Tips Coordinate geometry is a big focus on the ACT math section, and you’ll need to know its many facets in order to tackle the variety of coordinate geometry questions you’ll see on the test. Luckily, coordinate geometry is not difficult to visualize or wrap your head around once you know the basics. And we are here to walk you through them. There will usually be three questions on any given ACT that involve points alone, and another two to three questions that will involve lines and slopes and/or rotations, reflections, or translations. These topics are tested by about 10% of your ACT math questions, so it is a good idea to understand the ins and outs of coordinate geometry before you tackle the test. This article will be your complete guide to points and the building blocks for coordinate geometry: I will explain how to find and manipulate points, distances, and midpoints, and give you strategies for solving these types of questions on the ACT. 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 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 along the x and y-axes and its distance from 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, take look at the following graph. This point is 4 units to the right of the origin and 2 units above the origin. This means that our point is located at coordinates (4, 2). Anywhere to the right of the origin will have a positive x value. Anywhere left of the origin will have a negative x value. Anywhere vertically above the origin will have a positive y value. Anywhere vertically below the origin will have a negative y value. So, if we break up the coordinate plane 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 breakdown the different ways to solve these types of problems. May you always have fast vehicles (or at least sturdy shoes) for all your distance travel. 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 formula, or using the Pythagorean Theorem. Let’s look at both. Solving Method 1: Distance Formula If you prefer to use formulas on as many questions as you are able, then go ahead and memorize the distance formula above. You will not be provided any formulas on the ACT math section, including the distance formula, 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.) You will have to memorize each and every ACT math formula you'll need and, for those of you who want to learn as few as possible, the distance formula might be the straw that broke the camel’s back. But for those of you who like formulas and have an easy time memorizing them, adding in the distance formula to your repertoire might not be a problem. So how do we use our formula in action? Let us say we have two points, (-5, 3) and (1, -5), and we must find the distance between the two. If we simply plug our values into our distance formula, we get: $√{(x_2-x_1)^2+(y_2-y_1)^2}$ $√{(1-(-5))^2+(-5-3)^2}$ $√{(6)^2+(-8)^2}$ $√{(36+64)}$ $√100$ 10 The distance between our two points is 10. 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. Though, again, you won’t be given any formulas on the ACT math section, you will need to know the Pythagorean Theorem for many different types of questions, and it's a formula you’ve probably had experience using in your math classes in school. This means you will both need to know it for the test anyway, and you probably already do. So why can we use the Pythagorean Theorem to find the distance between points? Because the distance formula is actually derived from the Pythagorean Theorem (and we'll show you how in just a bit). The trade-off is that solving your distance questions this way takes slightly longer, but it also doesn’t require you to expend energy memorizing any more formulas than you absolutely need to and carries less risk of remembering the distance formula wrong. To use the Pythagorean Theorem to find a distance, 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 $(−5,3)$ and $(1,−5)$. 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 6 and 8. 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. $a^2+b^2=c^2$ $6^2+8^2=c^2$ $36+64=c^2$ $100=c^2$ $c=10$ The distance between our two points is, once again, 10. [Special Note: If you are familiar with your triangle shortcuts, you may have noticed that this triangle was what we call a 3-4-5 triangle multiplied by 2. 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 10 if the two legs are 6 and 8. 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, (-5, 3) and (1, -5). If we take the average of our x values, we get: ${-5+1}/2$ $-4/2$ 2 And if we take the average of our y values, we get: ${3+(-5)}/2$ $-2/2$ −1 The midpoint of the line will be at coordinates (−2,−1). If we look at our picture from earlier, we can see that this calculation makes sense. It is difficult to find the midpoint of a line without use of the formula, but thinking of it as finding the average of each axis value, rather than thinking of it as a formal formula, may make it easier to visualize and remember. So what kinds of point and distance questions are on your horizon? Let's take a look. Typical Point Questions Point questions on the ACT will generally fall into one of two categories: questions about how the coordinate plane works and midpoint or distance questions. Let’s look at each type. Coordinate Plane 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. This can take the form of testing whether or not you understand that the coordinate plane spans infinitely, or how well you understand how negative and positive x and y coordinate values will be, or how well you can visualize points and how they move within the coordinate plane. Let's take a look at an example: We know from our earlier chart that if x is positive and y is negative, then we will be in quadrant IV, and if x is negative and y is positive, we will be in quadrant II. Quadrant I will always have both positive x values and positive y values, and quadrant III will always have both negative x values and negative y values. These do not fit our criteria, so we can eliminate them. This means that our final answer is E, II or IV only. 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., â€Å"What is the distance from points (3, -5) and (4, 4)?†). Let’s look at an example of a scenario question, Becky, Lia, and Marian are friends who all live in the same neighborhood. Becky lives 5 miles north of Lia, and Marian lives 12 miles east of Lia. How many miles away do Becky and Marian live from each other? miles 12 miles 13 miles 14 miles 15 miles First, let's 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's continue on this path and simply use the Pythagorean theorem. Now, we can see that we have made a right triangle from the legs of distance we have already. Becky lives 5 miles north and Marian lives 12 miles east, which means that the legs of our triangle will be 5 and 12. Now we can find the hypotenuse by using the Pythagorean theorem. $5^2+12^2=c^2$ $25+144=c^2$ $169=c^2$ $c=√169$ $c=13$ [Note: if you remember your shortcuts for right triangles, you could have saved yourself some time and simply known that our distance/hypotenuse was 13. Why? Because a right triangle with legs of 5 and 12 means we have a 5-12-13 triangle, which means that the hypotenuse will always be 13.] The distance between Becky’s house and Marian’s house is 13 miles. Our final answer is C, 13 miles. On very rare occasions, you may also be asked for something slightly more peculiar on a midpoint or distance formula, such as the product or the sum of the coordinates. This just requires that you take an extra step once you’ve found your new coordinate points, so don’t get thrown by this scenario. We know that our midpoints are the averages of our individual coordinates. This means we can work backwards from our one pair of given coordinates and from our midpoint coordinates to find our second pair of original coordinates. Our first set of original coordinates is at (1,−5), so these will act as our $x_1$ and our $y_1$. And we are told that our midpoint is at (4,−3), so let us set up the problem. First, let us find the value of our $x_2$ (the x-coordinate of point B). ${x_1+x_2}/2=4$ ${1+x_2}/2=4$ $1+x_2=8$ $x_2=7$ Second, let us find the value of our $y_2$ (the y-coordinate of point B). ${y_1+y_2}/2=−3$ ${-5-y_2}/2=-3$ $−5+y_2=−6$ $y_2=−1$ Now we just need to add our two coordinates. $7+(−1)$ 6 Our final answer is C, 6. Now let's talk strategy, strategy, strategy. (Pretty sure saying things three times makes 'em lucky. Or just conjures Beetlejuice. Either way.) ACT Math 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 Your 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 given information. This is especially the case when working with negatives or with absolute values. In addition, most of the time when 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. 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 marking information on 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 resources at your disposal when taking the ACT and it takes little of each to make a rough sketch, but can cost you a lot more of both to keep all your information in your head. #3: Decide Now Which Formulas You Want to Use If you feel more comfortable using a variety of formulas for a variety of scenarios, then go ahead and memorize the distance formula in addition to all your other need-to-know formulas. But just remember that memorizing a formula wrong is worse than not remembering it at all, so make sure that you memorize and practice all your formula knowledge between now and test day so you can 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 more than a handful of formulas correctly on the day of the test), then go ahead and forget all your â€Å"optional† formulas. Take the time to memorize and use the Pythagorean theorem instead (since you’ll need to know it for a multitude of other types of problems anyway) and wash your hands of the rest of them. You’ll have to know at least a few formulas to do well on the ACT, but you can absolutely get by with only needing a handful, rather than needing to know them all. Test (about to be) in progress. Test Your Knowledge Now, let’s test your point knowledge on a few more real ACT math questions. 1. In the standard $(x,y)$ coordinate plane, a line segment has its endpoints at $(3,6)$ and $(9,4)$. What are the coordinates of the midpoint of the line segment? A. $(3,-1)$B. $(3,1)$C. $(6,2)$D. $(6,5)$E. $(12,10)$ 2. 3. 4. What is the distance between coordinates $(4, -2)$ and $(-4, -6)$? A. $4√5$B. $5√3$C. 8D. $9√3$E. 14 Answers: D, G, F, A Answer Explanations: 1. Here, we have a simple midpoint question, so we just need to find the averages of our coordinates. We are given $(3,6)$ and $(9,4)$, so let us first find the midpoint $x$-coordinate. $${3+9}/2=12/2=6$$ We know our answer must be C or D, since those are the only options that gives us our midpoint $x$-coordinate at 6. Now let us find our $y$-coordinate. $${6+4}/2=10/2=5$$ Our midpoint coordinates will be at (6,5). Our final answer is D, (6,5) 2. If we make a right triangle between the points we are given, we can see that it will have leg lengths of 8 and 8. Because the distance will be in proportion to the legs and the distance between E and D is $1/4$ the distance between E and F, we can take $1/4$ of the distance of each leg. So if we count 2 up from the $x$-coordinate and 2 up from the $y$-coordinate, we get a new coordinate point at (8,6). Our final answer is G, (8,6). 3. This is a question that may appear at first to be a beast to solve, but the principle behind it is not as complex as it looks. Once we've parsed the text, we can see that we are essentially just being asked to find the square root of the sum of the squares of our coordinate values ($√{x^2+y^2}$). The easiest way for us to do this is to plug in our own estimated values for our $z$ points. Because we are not given exact coordinate points, we know we will be able to solve the problem without exact coordinates, which means that a rough estimate will do just fine. So let's give each coordinate point a rough value and say that they are: $z_1=(−5,6$) $z_2=(−3,1)$ $z_3=(−3,−3)$ $z_4=(3,−2)$ $z_5=(5,2)$ Now we need to find the square root of the sum of the squares of our coordinate values ($√{x^2+y^2}$). This means that the squares will cancel out any negative coordinate values (because a negative times a negative is a positive). So we are just looking for whichever $z$ coordinate has the largest absolute value of its coordinates, and these would be $z_5$ and $z_1$. It looks as though $z_1$ will have the largest modulus value, but let's test them both just to be sure. $z_5$ $√{x^2+y^2}$ $√{5^2+2^2}$ $√{25+4}$ $√{29}$ 5.4 And $z_1$: $√{x^2+y^2}$ $√{(−5)^2+6^2}$ $√{25+36}$ $√{61}$ 7.8 The point with the greatest modulus value is $z_1$. Our final answer is F, $z_1$ 4. This is a typical distance question and we can, as always, either use the Pythagorean Theorem or the distance formula. In this case, let's just use the distance formula. $√{(x_2−x_1)^2+(y_2−y_1)^2}$ Our coordinates are: (4,−2) and (−4,−6), so let's plug that into our formula. $√{((−4)−4)^2+((−6)−(−2))^2}$ $√{(−8)^2+(−4)^2}$ $√{64+16}$ $√{80}$ $√16*√5$ $4√5$ (To understand how to reduce roots like this, check out our guide to advanced integers.) Our final answer is A, $4√5$ Oh yeah! You've earned some lasers! The Take-Aways The basic building blocks for coordinate geometry are understanding how the coordinate plane works and how points fit in and can be manipulated in it. Once you've grasped these fundamental concepts, you'll be able to perform more complex coordinate geometry tasks, such as finding slopes and rotating shapes. Coordinate geometry is not an insignificant ACT math topic, 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 ACT can throw at you. What’s Next? Want to brush up on any of your other math topics? Check out our individual math guides to get the walk-through on each and every topic on the ACT math test. Been procrastinating on your ACT studying? Learn how to overcome your desire to procrastinate and make a well-balanced study plan. Running out of time on the ACT math section? 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