CH 10 INTRO TO GEOMETRY

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1 59 CH 10 INTRO TO GEOMETRY Introduction G eo: Greek for earth, and metros: Greek for measure. These roots are the origin of the word geometry, which literally means earth measurement. The study of geometry has gone way beyond the notions of triangles and circles -- from the shapes of molecules to the structure of the 4-dimensional space-time continuum. Consider the following figure: First question: How many little squares are there in the figure? If you simply count them up, you ll get a total of 30 squares. Second question: Find the total distance around the figure. The total distance around consists of four pieces: the top, with length 6; the right, with length 5; the bottom, with length 6; and the left, with length 5. Adding the lengths of all four sides gives a total distance of 22. Notice that these two questions are very different. The first counts the number of squares inside the figure, while the second measures the distance all the way around the figure. Since the answers are different, we see that these two concepts are not the same.

2 60 Area The area of a geometric figure is a measure of how big a region is enclosed inside the figure. For example, the area of the region below is 18 square units. Just count the number of little squares, and you ve got the area. Perimeter The distance around a geometric figure is called its perimeter. For example, the perimeter of the region above is 20 units. Start at a corner (or anywhere you like), and march along the edge of the region until you reach your starting point. The distance you ve traveled (not the number of squares you walk by) is the perimeter. To help you remember this, note that peri means around, as in periscope or peripheral vision. Homework Find the area and perimeter of each geometric shape:

3 61 Rectangles and Squares In Manhattan, Fifth Avenue (which runs north-south) meets 42nd Street (which runs east-west) at a 90 angle. The floor and the wall also meet at a 90 angle. A rectangle is a four-sided figure with all inside angles equal to 90. This implies that adjacent sides (sides next to each other) are perpendicular and opposite sides are parallel. Notice that a square (where all four sides have the same length) is also a four-sided figure with all four inside angles equal to 90. Therefore, by definition, a square is a special kind of rectangle. We can conclude that every square is a rectangle, but certainly not every rectangle is a square. We know that the distance around the rectangle (the sum of all four of its sides) is called its perimeter. The area of a rectangle is a measure of the size of the region enclosed by the rectangle. From the first page of this chapter, we counted the little squares in the rectangle and got a total of 30. But notice that the rectangle could be viewed as having 5 rows with 6 little squares in each row. Multiplying 5 6 gives 30, the area. So for any rectangle (and therefore any square), Perimeter: Add the four sides Area: Multiply the number of rows by the number of little squares in each row (or, multiply the length by the width).

4 62 Homework 5. The length of a rectangle is 12 and its width is 5. Find the perimeter and the area of the rectangle. 6. Each side of a square is 10. Find the perimeter and the area of the square. Challenging Examples 1. The perimeter of a square is 44. What is the area of the square? To find the area of a square, we need to know the length of one side of the square, since all sides are equal. But the side isn t given; however, we can find it from the perimeter of 44. Since the perimeter is 44, it follows that each side must be 11. Now we can compute the area of the square by multiplying its length by its width (which are both 11 in this case). Since is 121, we conclude that the area of the square is The area of a square is 144. What is the square s perimeter? The perimeter of a square depends upon its side. We don t have the side, but we do have the area. Since the area is 144, and since the area is found by multiplying the length by the width (which must be the same in a square), we ask ourselves, What number times itself is 144? The answer is 12, so each side of the square is 12. Now, the perimeter of a square is the distance all the way around -- that is, all four sides added together. Adding four 12 s together gives a perimeter of 48.

5 63 3. The length of a rectangle is 5 more than its width. If the perimeter of the rectangle is 42, find the rectangle s area. This is a hard problem compared to the two previous ones, so we ll attack it by taking some educated guesses until all the conditions of the problem are met. The first requirement is that the length must be 5 more than the width. So consider the following rectangle: 11 6 I chose 6 for the width off the top of my head. Since 5 more than 6 is 11, the length would have to be 11. First, check that the length is actually 5 more than the width. It is, so this rectangle is possibly useful to us. Now see if the perimeter is 42, as it s supposed to be. Adding the four sides: = 34 We failed, so we ll try another (larger) rectangle. How about: 13 8 The length is still 5 more than the width. So we ll check this rectangle against the requirement that the perimeter be 42: = 42 This time we hit the perimeter right on. So we know that the dimensions of the rectangle we ve been seeking are 13 by 8. Since the problem asked for the rectangle s area, we multiply the length by the width to get an area of 104.

6 64 4. The length of a rectangle is 5 times its width. If the area of the rectangle is 80, find the rectangle s perimeter. This problem will also be done by taking some educated guesses until we land on the right answer. Let s randomly choose a rectangle whose width is 3. Since the length must be 5 times the width, we are considering the rectangle 15 3 This rectangle has an area of 45, nowhere near the 80 it s supposed to be. We ll increase the width to 5 and see what happens: 25 5 This rectangle has an area of 125, way too big. Let s compromise; since a width of 3 produced a rectangle too small, and a width of 5 produced one too big, let s hope that a width of 4 will do the trick The area of this rectangle is 80, exactly what was specified -- we ve found our rectangle. It should now be clear that the perimeter of this rectangle is 48. Homework 7. If the perimeter of a square is 8, find the area of the square. 8. If the area of a square is 25, find the perimeter of the square. 9. The length of a rectangle is 8 more than its width. If the perimeter of the rectangle is 40, what is the area of the rectangle?

7 The length of a rectangle is 2 times its width. If the perimeter of the rectangle is 30, what is the area of the rectangle? 11. The length of a rectangle is 2 times its width. If the area of the rectangle is 18, what is the perimeter of the rectangle? 12. The length of a rectangle is 6 more than its width. If the area of the rectangle is 72, what is the perimeter of the rectangle? 13. The length of a rectangle is 2 times its width. If the perimeter of the rectangle is 24, what is the area of the rectangle? 14. The length of a rectangle is 1 more than its width. If the area of the rectangle is 6, what is the perimeter of the rectangle? 15. The length of a rectangle is 4 times its width. If the area of the rectangle is 16, what is the perimeter of the rectangle? 16. The length of a rectangle is 8 more than its width. If the perimeter of the rectangle is 20, what is the area of the rectangle? 17. If the area of a square is 100, find the perimeter of the square. 18. The length of a rectangle is 5 times its width. If the area of the rectangle is 5, what is the perimeter of the rectangle? 19. The length of a rectangle is 1 more than its width. If the perimeter of the rectangle is 30, what is the area of the rectangle? 20. The length of a rectangle is 3 times its width. If the perimeter of the rectangle is 24, what is the area of the rectangle? 21. If the perimeter of a square is 28, find the area of the square. 22. The length of a rectangle is 8 more than its width. If the area of the rectangle is 84, what is the perimeter of the rectangle? 23. If the area of a square is 64, find the perimeter of the square. 24. The length of a rectangle is 4 times its width. If the perimeter of the rectangle is 40, what is the area of the rectangle? 25. If the area of a square is 144, find the perimeter of the square.

8 The length of a rectangle is 6 more than its width. If the perimeter of the rectangle is 28, what is the area of the rectangle? 27. The length of a rectangle is 2 times its width. If the area of the rectangle is 32, what is the perimeter of the rectangle? 28. If the perimeter of a square is 48, find the area of the square. 29. The length of a rectangle is 12 more than its width. If the area of the rectangle is 45, what is the perimeter of the rectangle? 30. If the perimeter of a square is 40, find the area of the square. 31. The length of a rectangle is 3 times its width. If the area of the rectangle is 27, what is the perimeter of the rectangle? 32. The length of a rectangle is 4 more than its width. If the perimeter of the rectangle is 20, what is the area of the rectangle? 33. The length of a rectangle is 10 more than its width. If the area of the rectangle is 39, what is the perimeter of the rectangle? 34. The length of a rectangle is 3 times its width. If the perimeter of the rectangle is 8, what is the area of the rectangle? Solutions 1. A = 38; P = A = 35; P = A = 9; P = A = 9; P = To find the perimeter, add the 4 sides: = 34 For the area, multiply the length by the width: 12(5) = To find the perimeter, add the 4 sides: = 40 For the area, multiply the length by the width: 10(10) = 100

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10 68 Each problem that I solved became a rule which served afterwards to solve other problems. Rene Descartes ( ), Discourse de la Methode