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Case Western Univ., summa cum laude
Perfect scorer on the SAT & the ACT
Devorah is the founder of Advantage Point Test Prep and the author of the book “Boost Your Score” The Unofficial Guide to the Real ACT.
Plane geometry, what you need to know. First Let's take a look at this game I'm playing, I'm playing darts right now and I'm actually really close to winning but to win I need to get six points or more in my next throw and now I'm curious, what are my odds, what are my chances of hitting somewhere within this inner circle that will give me six or more points in my next throw and actually you can solve this using geometry. So in this episode, we're going to first talk about plane geometry; things like circles you know triangles, squares and then we'll move on to coordinate geometry and talk about things like slopes and lines. Let's get started.
Let's take a look at the top plane geometry topics. Now you can see there are a lot of topics here but there's a lot of plane geometry on the ACT, there are 14 of these out of 60. So nearly a quarter of the test is plane geometry. We're going to go over all of these concepts because I want you to feel really really good about them before we head into the test. We're going to talk about lines and angles, triangles, circles, squares and rectangles and we're going to do a multiple figure problem, a problem that combines two or more shapes.
On to lines and angles, there's just a couple of things that I want you to remember. Now I'll just drive this out really quickly. Angles along a line add up to 180 degrees, you'll probably remember this from math class but just in case you know anytime you got your line, you've got two or more angles on it, these two angles here are going to add up to 180 so for example if this was 120, this angle here would be 60. Okay another thing; angles around a point add up to 360 degrees and vertical angles are congruent. So Let's say I had angles around a point, here's my point right here, so here we got angles all around a point, four angles all the angular measurements added together would equal to 360 degrees and by the way, vertical angles are congruent. Vertical angles are these angles on the opposite sides for example this is a vertical angle to this, so these angles will be equivalent to each other, also this angle is equivalent to this angle because it's also a vertical angle.
On to one of my favorite topics with lines and angles; when two parallel lines are cut by a transversal, some really neat things happen. You're probably thinking transversal, Oh my God, what is she talking about? All transversal is, is just a line cutting through two parallel lines, let me draw a picture and now you're going to say "Oh, you know what Devorah I've seen that before" Do you remember this? It's you know, you'll usually see oh this is line one, oh this is line two, they're parallel and you've got a line that goes through them. When you see something like this, there's some really cool things you can infer about the angles. As long as you have one angle measurement, you can find all of them. So for example Let's say I told you, this angle here is 60 degrees, great okay Let's find all the other angles. Well, we know because this is 60 that this has to be 120 because they're along the line like we talked about. Great and now we know that this one has to be 60 as well because it's vertical and this one has to be 120 as well because it's a vertical and also because they're along the line with the other angles. Okay so this 60 here, this is 120 here. Okay but what about this bottom part? How can you figure out what those are? This is where another rule comes in, when you have two parallel lines cut by a transversal, alternate, interior angles are congruent. Those are the angles that I say you know, are kind of inside the Z that you see right here so this 60 degree angle right here is going to be equivalent to this angle right here, you see these opposite internal angles to these two parallel lines and the same thing here because this angle here is 120, this angle here opposite interior is going to be 120, so this is 120 and then this one here is going to be 60. Okay so hopefully you can see that we've got 120 and 60 here and 60 and 120 here and then once you've got this you know again this is a vertical so this is also 120 and this is a vertical so it's also 60 and there we go. So just from knowing one angle when you have two parallel lines cut by a transversal, you can find all the angles.
Let's move on to our next big topic, triangles. You'll usually see three or four triangle problems on the ACT so you want to feel pretty comfortable with triangles. Let's have a look at the first thing that they'll ask; area of the triangle, you need to know this formula. Remember on the ACT, they don't give you any formulas, you really need to know them. Area of a triangle is base times height divided by two. Let me give you some examples of how to find an area. The easiest ones are right triangles, Let's say I'm going to draw a right triangle here, there we go I'll put [in my little right angle and Let's say I tell you, okay you know this is four, this is five, okay easy to find the area. We've got our base, we've got our height, if you're ever not sure what's the base, what's the height, the base is at the bottom, the rule is that to be a height, you have to hit the base at 90 degrees so that's the definition of a height; it's hitting the base at 90. Here we go so that's easy, you got your height and then your base and you know the area will be 4 times 5 which is 20 over 2 which is 10; so area would be 20, 4 times 5 over 2 which would be 10. Pretty straight forward but, what happens when you don't see a height that stands out to you, that's hitting at a 90 degrees. What if you see a triangle instead that looks something like this and you look at it and you're thinking what's the height, what's dropping down? Hopefully you'll remember this from school, this is when you will drop an altitude, right so you will see this kind of dotted like almost fake line that you're drawing just to find the area dropping down from the bottom and now this is your base, this whole bottom and this is your height and on the ACT, they'll always be a way for you to figure out what that length is going to be so that you can find the area. So again these are the two major different permutations of how you're going to look for area. Always look for your base and your height, if it's not apparent because it's not a right triangle, make sure you're looking for a place you can drop an altitude and then you'll have your height and your base and once again it's going to be just whatever the base is times the height divided by two.
Let's move on, more about triangles, perimeter we know perimeter is everything added up, all the sides so it will be all three sides added together is the perimeter of the triangle. All three angles of a triangle add up to 180 degrees, that's another thing that is tested occasionally with triangles. So all your three angles adding up to 180. On to another common triangle concept, similar triangles. Similar triangles have equal angles and proportional sides, I like to say they're kind of like Doctor Evil and Mini me. One of the triangles will kind of be a miniature version of the second one, so let me show you what I mean; you'll see something like this, you'll have a big triangle and a little triangle and then you'll know that the angles are congruent, maybe they'll just be lines like these [IB] you know oh, this angle's the same as this one and this angle is the same as this one which means that the third angles also have to be congruent. In a case like this, you know okay if the angles are the same, the size have to be in proportion. So Let's say you know, that this one has a side of two and this one has a side of four. Proportion. I know that all these sides are going to be double, all the sides of this triangle so for example this one's you know two, three, four, this would be something like four, six and eight. So you see the sides are proportional and again we talked about proportions in the pre-algebra episode, there will be times in the ACT where you're going to actually have to find one of the missing sides using proportions and for more practice just head to your bonus materials, I've got some [IB] to practice this.
Let's move on to right triangles, this is the most common triangle you'll see on the ACT. So for right triangles, a few things you need to know. First one angle is equal to 90 degrees that's a right triangle, second you got the Pythagorean Theorem to know; a squared plus b squared equals c squared, you need to know this so memorize it if you don't know it by heart already. Let me just show you a quick example of using the Pythagorean Theorem; if I had a right triangle right here and I knew for example that this side was three and this side was four and I was looking for the hypotenuse, this side opposite my right angle and we don't know what it is. Okay we know that 3 squared plus 4 squared is going to give us X squared so we have 3 squared plus four squared equals X squared. Okay, so we know that nine plus 16 is X squared. So 25 equals X squared and so therefore you are going to take the square root of both sides to remove your exponent here and X is the square root of 25 which is equal to five. So here we go, our third side here would be five on our right triangle. That's an example of using the Pythagorean Theorem. And we'll have another example later to practice again.
Some other things with right triangles, 45-45-90 and 30-60-90 triangles are the most commonly tested kind of right triangles. What this means is that you'll see triangles that either have angles that are 45 degrees, 45 degrees and 90 or 30, 60 and 90. And when you see triangles like these they have very specific side of proportions always. So let's start with the 45-45-90 and I'll show you what I mean. So if I have got a 45-45-90 triangle here, okay so we know 45-45-90, there's some specifics that are always the same. Always you'll have the sides in the following ratio: you're going to have X-X-X square root of two and you're thinking, what is she talking about? I'll show you. What this means is that the side opposite the 45 degree angles, we'll call those X, the short sides. So we are going to have these we call those X. The side opposite the 90 degree angle, your hypotenuse, is going to be radical two times the shortest side. Well here we've got two short sides so radical two times either of the sides always. So that's your proportion, 45-45-90, you've got X if you're going to call the side opposite of 45 X; you've got X-X-X radical two.
Let me put it into practice so you see what I'm talking about. If I told you for example, one of the sides was say three. Well, it's a side opposite of 45 degree angle, that's our X, that's our shortest side for the proportion so we know from this proportion that always the side opposite the 90 degrees is radical two times the shortest length. So now I know even if I'm doing the Pythagorean Theorem, that this hypotenuse right here, the side opposite the 90 degree angle is radical two times three so three radical two and also because these are both opposite 45 degree angles, these are equivalent so both of these sides are going to be three. So that's how side proportions will work, on 45-45-90.
Let's take a look at 30-60-90. So in this situation, I'll draw another triangle here, the proportion is a little bit different. Here's your 30, here's your 60. Okay, here your proportions are the following. If we call the shortest side X, so the shortest side is always going to be opposite the smallest angle. So if we have our short side X opposite the 30. So you're going to have the side opposite the 60 degree angle is always going to be radical three times X, so X radical three and the hypotenuse is always twice the shortest side. So just for proportion, I'll put two X so if X is your shortest side, your middle side is radical three times that and your hypotenuse is twice that. So let's see how it would look with real numbers. If I told you for example, that opposite your shorter side, you had something like four. You would know okay, well then opposite my 60 degree angle; I've got four radical three, right? My short side times radical three. And I would know right away that my hypotenuse was twice that short side, which is eight. Now on the ACT sometimes when they get a little more complicated, they won't give you the shortest side, so you'll have to work backward. For example, maybe they'll give you eight and then you have to remember huh! Think proportions, oh! The hypotenuse is always twice the short side so if hypotenuse is eight and I work backwards, the short side must be four and then once I know the short side is four, oh that middle side, radical three times the short side so four radical three.
So that's how you would use these proportions. I know it sounds really complicated and students usually need some practice before they fully understand this, so that's why there's a nice tutorial in your bonus materials. If you're feeling a little iffy and you don't really remember from when you learned this in high school, head to that tutorial and hopefully you'll be feeling really good about this.
Okay let's look at an example problem where we'll put some of these concepts into play. Here we go, we've got our diagram here, it says figure not drawn to scale by the way we know on the ACT we talked about in our strategies, usually they are but in the figure above, BC is radical two, what's the length of AD? This is kind of a difficult problem. You don't have a lot of information, you've got a side length over here in this triangle and they are asking for the side length over here in this triangle. But what do we see that we can use? We just talked about right triangles and we just talked about the really common ones, the 45-45-90 and the 30-60-90. Here we go, we've got a 45-45-90, right? We know this side has to be 45 because we said angles add up to 180 degrees, you've got a 90 degree, you've got a 45, this is going to have to be 45, alright then. So now what can we do with this information? We've talked about the side ratios right? We know that if you have the two sides opposite the 45 degree angles, if we call those X or whatever we call them, you're going to have the hypotenuse be radical two times those lengths, either one of them. So radical two times either one of the lengths. So here even without doing the Pythagorean Theorem, which we could do. We could do you know, radical two squared plus radical two squared because these are equivalent, equals our hypotenuse squared but a lot faster to just think, I know hypotenuse radical two times my shortest length because it's a 45-45-90 so radical two times radical two which would just be two. So here we go, I can draw on this two and now we have a piece of information that will help us with this triangle which looks suspiciously like a 30-60-90 triangle. And keep a lookout because again this will show up with the most frequency. So seems a little disguised but now we look closely, we're missing the 30 we drew in, it's a 30-60-90 and again side proportions that can help us. What did we talk about? We talked about how in a 30-60-90 the hypotenuse, the side opposite the 90 degrees is always twice your short length, twice the shortest length. The shortest length again opposite the smallest angle, that makes sense so here because we know that side, that side is two, then we know that the hypotenuse is twice that which would be four so here we go. We've got AD, we've got our hypotenuse and that's the answer. So the answer choice here would be D.
Let's keep going and talk about circles. So you learned this in basic geometry but just to review. Here's a circle and you've got equations for your area and your circumference. The equation for area is pi R squared and the equation for circumference is two pi R. Now if you're thinking, what's R again? R is the radius; remember a radius is the line that goes from the middle of your circle to the outside. So this for example would be your radius. Diameter is twice the radius or the length all the way across the circle. And just a quick reminder. Sometimes students ask me what's the difference between area and circumference, I tell then for area think a carpet, you're covering a surface area. That would be the whole inside, on the other hand, circumference that's like perimeter, that's like a fence and that would be measuring just the outside line, how long that is.
Let's take a look at a problem, back to my dart board. You remember I was wondering what are my odds of getting six or more points on my next throw. Let's take a close look. To get six or more points I have to hit in this inner circle anywhere from over here where we've got six, seven, eight or nine point options if I hit the middle, that's where I need to hit. How am I going to find out what my odds are? Well, let's think about it, I can find out what my odds are by figuring out what's the area of that inner circle where I want to target versus what's the area of the whole board which I might be hitting. So let's figure that out, I measured it and I found that the radius of my whole dart board is six inches and the radius of my target area is four inches. Okay, let's find the area of both. Target area first. So we have got the radius of my target area is four and remember area pi R squared, okay so pi R squared, so pi times 4 squared so 16 pi. 16 pi is my area for the little part that I want to hit, what's the area of the whole thing? Well we know the radius is 6 so again pi times 6 squared okay. So the area of the whole thing, 36 pi. Now I've got the areas of both right? Area of the small one 16 pi, large 36 pi so what's the odds? Well the odds are just 16 pi over 36 pi right? And so let's figure that out. What's 16 over 36? I'm going to use my calculator. So 16 divided by 36 is about point four four, so 44 percent. So there's actually a 44 percent chance of me hitting this middle part out of this whole circle on my next throw, pretty cool. Let's take a look at some more geometry.
Multiple figure problems, these are really tough, you'll see a couple of them and what this require you to do is actually combine what you know about a couple different shapes. So here, this is one of the hardest problems you would ever see on the math ACT. In the figure above, a square is inscribed in a circle with an area of 16pi. What is the area of the square? Okay, hopefully you'll remember area of the square, length times width right? A side times a side, so we're going to find the sides and we kind of have a hint that we're going to need the circle to help us. Well, we know that this circle has an area of 16pi, remember area pi R squared so what squared gives us 16? 4 . Okay so we know that the radius is 4, so if the radius the part from here to here is 4, what else can we do here? We know that the diameter is twice the radius right? So from here to here across this square and across this circle, that's going to have to be 8 right, it is that whole length. So now we know and we write diameter equals 8. Alright so hopefully you're following me, this again is a really tough problem. We've got the radius is 4 therefore we know the diameter across the circle is 8 and oh, by the way this helps us with our square now, because now we have a line across the middle of our square that's dividing it into two right triangles, two 90 degree triangles and we can use something we learned when we talked about right triangles here so see another piece of information about a shape. Okay well we talked about how you usually have 45-45-90 or 30-60-90 right triangles on this test.
Well now that you've got this shape the square with four 90 degree angles and you're chopping these angles in half, you end up with two 45-45-90 triangles right? Okay so stay with me we're almost done. Two 45-45-90 triangles in that case what did we talk about proportions right? We talked about how in a 45-45-90 triangle the hypotenuse, the side opposite the 90 degrees is radical two times one of the legs, this is going to help us. We know that 8, the length of this whole diagonal here is radical two times one of these sides and again these sides are what we need to find the area, okay. So Let's do this out, so we know that 8 equals radical 2 times the side okay so Let's just simplify and then we can figure out what our side is, we know if we move the radical 2 over, 8 over radical 2 equals a side and now we're just going to multiply the sides to get our area. By the way you might be thinking why is the radical in denominator? Do we need to change that and hopefully you learned in school about rationalizing the denominator, here we don't care about it because we're going to be multiplying two radicals at the bottom so this is going to turn into finding the area now or two sides so sides times side is 8 radical 2 times 8 over radical two and then you end up with 8 times 8 is 64 over two because these radicals cancel out. So you've got 64 over 2 and now let's simplify to 32, great and that's answer choice D and again this is really the hardest type of geometry problem you would see so if you got it, feel great, if not go over it again, okay. That's it for top plane geometry concepts you need to know, remember this make up a quarter of the test so you want to feel really, really good about them and so if you want more practice make sure to head to your bonus material.
Coordinate geometry, there are nine of these on ACT math and we're going to look at the top three concepts that are going to be tested. First equation of a line, slope and finding intercepts. Equation of a line, hopefully you're remember this from school, y=mx+b Let's just take a look at the parts. So m is the slope and we'll talk about that more in a second, and b is the y intercept. We're hitting along the y axis and x and y in the equation usually stay x and y but actually what they really are, are coordinates of any point along that line. Let's take a look at slope; slope is rise over run, the change in y for every change in x. All that really means is that when you see a line, slope would be how much it goes up over how much it goes over. So here, we see we're going up 1 and then over 1 so our slope would be 1 over 1 or just 1. Let's take a look at the equation; on the ACT there will be times when you have two sets of points and then you've got to find the slope for those. This is the equation; Y2 minus Y1 over X2 minus X1, Let's take an example so you see what I mean. Find the slope of a line passing through the points 3,1 and the point 4,2. Okay great, remember our equation, Y2 minus Y1 over X2 minus X1 so our second Y which is 2 minus our first Y which is 1 over our second X which is 4 minus our first X which is 3 so what do you get? You get 1 over 1, just 1. So our slope here again is just 1, answer choice D.
Moving on, let's talk about finding intercepts. So I like to remind students the difference between the X and the Y intercept because there are questions about both on the test. The X intercept is defined as the X coordinate of the point where the line hits the X axis so let me show you, if I have just a little coordinate plane over here. You see how when we hit along the X axis, that will be right over here and when you hit along the X axis that's your X intercept and in that point your Y is going to actually be zero. You're hitting along the X, X is going to have a number, the Y coordinate though, you're not going anywhere in y and that would be zero. Same thing with Y, if you have ever seen Y intercept, that's the spot where you hit along the Y over here, and at that point X is going to be zero. Let's take a look at an example; What is the Y intercept of the line y minus 4x is equal to negative 9. So a couple of different ways to do this, you might look at this and think this line looks kind of funky, this equation, well that's just because it's not in a y=mx+b form, so what you could do is rearrange it so it's in y=mx+b form and then you'll remember b is your Y intercept and you can find it that way. You also can remember well at my Y intercept, my X is equals to zero and just set X to zero so you know, if you had a y minus 4 times zero, that would just be y minus zero and that would equal negative 9. So here we go, our y intercept is just negative 9 and that's it, we've finished our review of top coordinate geometry concepts, remember if you need more help there's a lot in your bonus materials for you to practice with.
Let's recap, we talked about plane geometry and we went over the top concepts you need to know. Remember there are 14 out of all the 60 math questions that are going to be about plane geometry so make sure you're comfortable with these concepts. We also talked about coordinate geometry, and remember these will show up nine times so you see there's a lot of geometry on ACT math. Make sure you really practice so you feel great about these concepts on test day.
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