1 00:00:02,250 --> 00:00:03,030 Robert McKeown: We've been given some 2 00:00:03,030 --> 00:00:07,320 information about a rectangle, we know that the 3 00:00:07,350 --> 00:00:12,390 perimeter of this rectangle is 112 units. And 4 00:00:12,390 --> 00:00:16,920 we're being asked to find the length of side v 5 00:00:16,920 --> 00:00:24,090 w. So this side right here, it says, write your 6 00:00:24,090 --> 00:00:28,620 answer without variables. So we need to solve 7 00:00:28,620 --> 00:00:38,010 for, we need to solve for y. Now, the perimeter 8 00:00:41,370 --> 00:00:51,090 is equal to two times the width of the 9 00:00:51,120 --> 00:00:55,110 rectangle, multiplied by the length of the 10 00:00:55,110 --> 00:01:01,140 rectangle. a rectangle has two sides with the 11 00:01:01,140 --> 00:01:05,100 same width two sides with the same length. Now 12 00:01:06,300 --> 00:01:10,110 Oh, excuse me, I made a mistake. I think I said 13 00:01:10,110 --> 00:01:12,180 it correctly, but I didn't write it correctly. 14 00:01:12,180 --> 00:01:16,380 And that's two times out. So we've got two 15 00:01:16,380 --> 00:01:19,350 widths plus two length is is equal to the 16 00:01:19,350 --> 00:01:26,730 perimeter of the rectangle. Now we know that the 17 00:01:26,730 --> 00:01:32,910 perimeter is equal to 112. And we know that the 18 00:01:32,940 --> 00:01:42,120 side x w is equal to five y minus one. And we 19 00:01:42,120 --> 00:01:49,380 know that the side v w is equal to four y plus 20 00:01:49,380 --> 00:01:55,080 three. Now we're going to multiply out the 21 00:01:55,080 --> 00:02:01,440 brackets. And we end up with 10 y minus two plus 22 00:02:01,470 --> 00:02:06,750 eight y plus six, and that whole thing is equal 23 00:02:06,750 --> 00:02:12,090 to 112. Now combining like terms, could I have 24 00:02:12,120 --> 00:02:18,030 18 one plus four. And that's like minus four on 25 00:02:18,060 --> 00:02:22,410 both sides, we've got 108 is equal to 18. Why? 26 00:02:25,290 --> 00:02:27,810 Now I know that y is going to end up being equal 27 00:02:27,810 --> 00:02:34,530 to 108 over 18. What can we do with that? Well, 28 00:02:34,590 --> 00:02:37,470 they're both even numerator and denominator. So 29 00:02:37,470 --> 00:02:40,440 I'm just going to divide both by two. Now I've 30 00:02:40,440 --> 00:02:44,910 got 54 over nine. And when I divide 54, over 31 00:02:44,910 --> 00:02:48,870 nine, I noticed that 54 is equal to nine times 32 00:02:48,870 --> 00:02:52,320 six. And so I can rewrite this as six. So we've 33 00:02:52,320 --> 00:03:03,180 got y is equal to six. Now, what's the length v 34 00:03:03,180 --> 00:03:10,530 w, it's going to be equal to four y plus three, 35 00:03:10,920 --> 00:03:13,920 which means it must be equal to four times six, 36 00:03:14,670 --> 00:03:20,100 plus three, which means it's equal to 24 plus 37 00:03:20,100 --> 00:03:25,380 three or 27. Saw the length of the side, v w is 38 00:03:25,380 --> 00:03:29,730 equal to 27. Now let's go into ALEKS and see if 39 00:03:29,730 --> 00:03:33,180 we've got the right answer. Here we are in 40 00:03:33,180 --> 00:03:35,460 ALEKS. Here's the question we were looking at. 41 00:03:36,090 --> 00:03:43,740 All I have to do is type in 27. And I will click 42 00:03:43,740 --> 00:03:47,640 the check button. And we got the right answer. 43 00:03:47,940 --> 00:03:51,030 So we just have to remember what a rectangle the 44 00:03:51,030 --> 00:03:55,110 properties of a rectangle and we can use, 45 00:03:55,320 --> 00:03:58,800 manipulate the equation to find out the length 46 00:03:58,800 --> 00:03:59,520 of one side. 47 00:04:07,200 --> 00:04:10,080 Pythagorean theorem. So we do use this in 48 00:04:10,080 --> 00:04:12,840 economics, we often use this and undergraduate 49 00:04:12,840 --> 00:04:17,610 economics when we're trying to simplify social 50 00:04:17,610 --> 00:04:21,450 welfare calculations. Won't you probably have no 51 00:04:21,450 --> 00:04:23,340 idea what that means. But you're probably going 52 00:04:23,340 --> 00:04:26,730 to come across the Pythagorean theorem in your 53 00:04:26,730 --> 00:04:29,460 micro economics class, also, possibly in your 54 00:04:29,460 --> 00:04:32,850 econometrics class, where you can use a 55 00:04:32,850 --> 00:04:36,720 geometric approach to statistical analysis. So 56 00:04:36,720 --> 00:04:40,050 let's take a look what is Pythagoras theorem, it 57 00:04:40,050 --> 00:04:45,360 really just says that the first short side of 58 00:04:45,360 --> 00:04:54,360 the triangle squared plus the other short side 59 00:04:56,340 --> 00:05:03,510 squared is equal to The long side or the high 60 00:05:03,510 --> 00:05:08,370 pot news square. So an easier way to remember it 61 00:05:08,370 --> 00:05:10,380 is the short side squared plus the short side 62 00:05:10,380 --> 00:05:16,500 squared is equal to the long side squared. Now 63 00:05:16,500 --> 00:05:20,820 let's take a look at a problem. So we've got a 64 00:05:20,820 --> 00:05:23,490 Pythagoras theorem here, we've got a right 65 00:05:23,520 --> 00:05:26,700 triangle, and we're being asked to find find the 66 00:05:26,700 --> 00:05:32,610 side length x. So we know that the long side 67 00:05:32,850 --> 00:05:36,690 squared is equal to the short side squared plus 68 00:05:36,690 --> 00:05:43,230 the other short side squared, multiply this out, 69 00:05:43,260 --> 00:05:48,630 we get 100 is equal to 64 plus x square x 70 00:05:48,630 --> 00:05:52,950 squared is equal to 36. And x is equal to six. 71 00:05:59,550 --> 00:06:02,700 Here's another question with triangles. So we're 72 00:06:02,700 --> 00:06:05,130 being asked to find the area of the triangle 73 00:06:05,130 --> 00:06:13,950 below. Well, if I draw a triangle, like that, 74 00:06:14,580 --> 00:06:15,510 notice 75 00:06:23,670 --> 00:06:28,590 that it's half the area of a rectangle, it's 76 00:06:28,590 --> 00:06:33,000 half the area of a rectangle. And so the formula 77 00:06:33,000 --> 00:06:37,920 here for the area of a triangle is just one half 78 00:06:38,430 --> 00:06:46,680 times the base times the height. Or similarly to 79 00:06:46,680 --> 00:06:52,740 with Pythagoras theorem, it's one half the first 80 00:06:52,740 --> 00:07:00,690 short side times the second short side, and we 81 00:07:00,690 --> 00:07:04,020 don't need the high pot news to get the area of 82 00:07:04,020 --> 00:07:06,960 the rectangle. So what is the area this 83 00:07:06,960 --> 00:07:09,390 rectangle? Well, it's going to be one half, 84 00:07:10,110 --> 00:07:17,370 times 12 times 16. And if I want to calculate 85 00:07:17,370 --> 00:07:19,710 this without a calculator, I could rewrite it as 86 00:07:19,710 --> 00:07:23,970 12 times eight, make the largest number a little 87 00:07:23,970 --> 00:07:27,360 bit smaller. And if I wanted to fully factor 88 00:07:27,360 --> 00:07:31,230 this thing out, I could have three times four, 89 00:07:31,650 --> 00:07:39,960 times four, times two. And maybe I'll go three 90 00:07:39,960 --> 00:07:44,610 times four, that gives me 12. For 12, sets 4848 91 00:07:44,610 --> 00:07:48,630 times to the area, this rectangle is 96. 92 00:07:56,010 --> 00:07:59,010 Our last problem, we'll look at the angle 93 00:07:59,010 --> 00:08:02,250 measure of a triangle, and very simply all 94 00:08:02,250 --> 00:08:16,800 triangles. Angles sum to 180 degrees. And so 95 00:08:16,800 --> 00:08:19,620 that makes answering this problem really quite 96 00:08:19,650 --> 00:08:25,800 simple. We want to know what this x value is the 97 00:08:25,800 --> 00:08:28,770 angle of this corner up here, it's going to be 98 00:08:28,800 --> 00:08:36,780 equal to x plus 68 degrees plus 30 degrees, it's 99 00:08:36,780 --> 00:08:41,310 going to be equal to 180 degrees. And so x is 100 00:08:41,310 --> 00:08:47,910 just going to be equal to 82 degrees. Angles not 101 00:08:47,910 --> 00:08:50,340 something we use very much in economics. But if 102 00:08:50,340 --> 00:08:52,470 you're interested, here's a start for a with 103 00:08:52,470 --> 00:08:55,830 some other problems in the ALEKS geometry 104 00:08:56,280 --> 00:08:56,910 module.