Unit 4 Lesson 2 Practice Problems IM® GeometryTM authored by Illustrative Mathematics®

Number one wants us to find the legs of this right triangle shown. And so when we're looking at this right triangle, hopefully you notice that not obviously it's right triangle, but then also one of the angles is 45°. And so what we learned is if we have right triangle and one of the other angles is 45 then that leaves 45° left over for this other angle as well. and then that makes it an isosles right triangle. So these legs right here are the same length as well. So this allowed us to know that this is similar to every other 45 4590 right triangle. And so if you kind of take look at one that call the parent right triangle, the parent 4590 triangle. This one has legs of 1 1 and 2. So this is the one that all other triangles all other 45 90 triangles are similar to. And so if we take look at this with the scale factor and you can kind of do it between the sides. So if you have the side length to get to the hypotenuse, we can see that we just multiplied one by the of two. Now if we had the hypotenuse, so if we knew the hypotenuse, then we take it and we divide by the square root of two to get to the leg. And this will work in every 45 45 90 right triangle. And so in this one we have the hypotenuse given to us. So if we take it and we divide by 2 we'll get the length of this leg. So let me just call the leg here. and so is going to equal 4 / the of two. And so in our answers here, we see that that's actually written as fraction in part Number two wants us to come up with the length of the diagonal of this square. So when we cut squares when we cut square in half with its diagonal, it creates two 4590 triangles. Since we've got the legs the same, that forces these two angles to be the same, which would force them to be 45°. And so now, if we want to go find the length of the diagonal, now we have the length of one of the legs. And so when you have the leg, you multip. So if just call this hypotenuse is going to equal the leg one times that square two. So is just going to equal of two. You can also if you like better you can use that parent triangle idea. So let me just draw this parent triangle over here also and give you one other strategy. So if we have this triangle and you can remember on 45 45 90 that that means that the legs are ones and the hypotenuse is square of two. Then you can also set up proportions. mean and if you recognized this, this one just has one and one over here. And so then that would have to be square of two. but you can also match the corresponding sides. So this one's one. So would put one underneath this one. And then the hypotenuse is square 2. So would put square two under this And then set those equal to each other. So 1 over 1 = over 2. And then you'd be able to cross multiply and you'd get 2= that way. So you can also set up proportions with similar figures if you'd rather. Number three, square has diagonal with length of 5 cm. What is the area of the square? So, we know this diagonal here is five. We also know that the legs are all the same in square. So, we have two triangles. And so we just need to find one of these legs and then we'll know both in order to do the area. So area formula for square is just to take one of the sides and square it cuz it's just base time height and the sides are going to be the same. So let's take look again. if you want to be looking at that parent triangle to help you, we've got each of these side lengths being one and then the hypotenuse being square of two. So remember, in this one, we have the hypotenuse. And so to get to the leg, we would just divide by 2 if you want to be doing that. So is just going to equal 5 / the of two. And so that's what each of these sides are. And so then we're going to do the area formula. And the area formula is going to take the side 5 two and it's going to square it or multiply it by itself. So in this case 5^ squar on top is 25. And then the square of 2^ squared is just plain two. So it looks like this. Square of two squared. those will just cancel and you'll get two. And so you can leave it like that. or you could do 25 / 2 and get 12.5 units squared or centime squared in this case cuz they actually told us that it was 5 cm. and then if you want to do that proportion method, this is maybe better one to show you. So this leg is one. So on the leg I'm going to put that over one. And then the hypotenuse is square 2. So I'm going to match that 2 with the five. So then you would have gotten over 1 = 5 over 2. And we know anything divided by itself doesn't change it. So then you would have ended up with 5 over 2 for the side length just like we did this way. Squared it for the area. Number four. Pria is teaching her younger cousin to ride bike. She wants to stay on roads that are not too steep and that they're easy enough for new biker to ride. So, she's decided that roads must have an angle less than or equal to 7°. And she knows that 7° angle in right triangle has 3 to 25 ratio for its legs. So, list the legs of two right triangles that would be safe for new biker to ride. So, I'm just going to draw out 3x 25. So if the height is three then the leg would be 25. So it would be something like this. And now in order to keep this angle smaller either you could just leave the 25 the same and then the height just needs to get shorter so that that angle decreases. So you could do something like 2x 25. So that's certainly going to get smaller angle. So, 2x 25. or we could leave the height the same and just extend this length. So, if we left the height at three, then we would just need to extend this further. That's going to pull that angle to be smaller angle as well. So, you could leave this at three. And then you could put this leg at something like 26 or higher. So, that will kind of guarantee it. if you wanted to change these measures and then check just make sure you look at 3 divided by 25 so that decimal which is 0.12 and then you compared your height divided by your length and you just make sure that that's less than or equal to that.12. Number five, Claire and Han are discussing how to find missing lengths. Claire says she's going to use similarity and Han says he's using Pythagorean theorem. Do you agree with either of them? And show your reasoning. and so they're going to find the missing lengths of both of these. So for Pythagorean, we have to have right triangle. And we do have right triangle in both cases with two of the legs given to us. so here's the right angle. and then two of the legs are given here. So Pythagorean theorem is certainly going to work. So will do that on triangle DF. So we're given the hypotenuse is 13. So 13^2 = 5^2 + d^ 2. So 169 = 25 + d^ 2. subtract 25 and we get 144 is equal to ED^2AR and then we'll square root and we get 12 for ED. So that is totally fine way of doing it and you could certainly do that with this one as well. Do 2.5^2 + 6^2 = AC^2. and then we can in fact also do similarity because they share two angles. So we've got right angle that's equal to each other and we've got this angle marked as congruent. So we know that the two triangles are similar. So we're going to be able to use similarity as well. So this height kind of here, this short leg 2.5 goes with the five. And then this longer leg six goes with this longer leg of 12. And so that shows us that we actually have scale. So now if I'm going to go 13 back to this way, want to Whoops. want to look at taking the 12. How do we go from 12 back to six? And so you can think of that as multiplying by 1/2 or dividing by two. So can take this 13 divide it by two and that will give me this side of 6.5. So either way that you do this you could do similarity or Pythagorean theorem. So agree with both of them. Number six in right triangle ABC angle is the right angle. So let's draw out picture of right triangle here. So angle is the right angle and then you can label and whichever other ones you want and it says that AB is 25 units long and BC is 24 units long. What is the length of AC? So since it's right triangle with two legs given we know we can do Pythagorean theorem. So we square the hypotenuse and then we square both of the legs and add them together. 25 squar is 625. 24 squar is 576 and then we have that AC^ squar. Subtract the 576 from both sides and you get 60. no what do we get? 49. Okay, so 49 and then we have that AC squared and then we will square root both sides and get seven for the length of that missing side. Number seven, find the length of EF. So we're looking for this side length. And so we got to make sure that these two triangles are similar before we do anything since we only see this 55° angle in common. So let's first do 180 - 55 - 82 and find out that we get 43. So this angle here is equal to 43 which we also see here. So we know for sure that we can do similarity. And then let's kind of match up some of these sides. So EF is the side that connects 43 to the 82° angle. And so we can mark that here. The 43 to 82 is BC. Then we also see this 21 which connects the 43 and the 55. And over here we see the 14 is the side that connects the 43 and the 25. And so those sides are what we're going to use to set up proportion or you can do scale factor if you would like. So I'm just going to do proportions in this one. So EF. So I'm going to compare side to 12. And then we can compare 21 to 14. And then we'll be able to cross multiply here. And so we'll do 12 * 21 which is 252 and then we will divide that by 14 and we will get 18. So EF equals 18. And again, you could have used skill factor here as well. So you could have looked at what do we do to the 14 to get to 21? So that's 21 / 14, which if you wanted to simplify those both divide by 7. and so that would simplify down to 3 over2. So then this EF is going to be 12 * the scale factor. So we're going to do 12 * three halves or time 1.5 and that's going to give us 18 that way as well. And then angle actually already wrote in and that is 82°. Number eight, determine if each statement must be true or could possibly be true. Isosesles triangles are similar. so we certainly can get some isosesles triangles that are similar. So if did this, here's an isosles triangle. could mark these sides the same. And if just grouped these on my page and then duplicated it and made it bigger. These are going to be similar to each other. just dilated it. So certainly we can have isosesles triangles that are similar. But all isosesles triangles don't have to be because could draw this one and this one is certainly not similar to these ones because this is an obtuse or much bigger angle than this one. So this could be true but does not have to be true for equilateral triangles. Okay, equilateral triangles always have to be similar. And that's because all three sides are the same. So no matter what these side lengths are, if call all of these then if do another equilateral, whoops, another equilateral triangle, could name all of these sides and they're all equal to each other. So if divide all these sides down to check if the sides are in proportion, I'm going to have over every single time. So they are all going to be in proportion to each other. So by side side side similarity that would always have to be true.