Rectangle Square Triangle


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1 HFCC Math Lab Beginning Algebra  15 PERIMETER WORD PROBLEMS The perimeter of a plane geometric figure is the sum of the lengths of its sides. In this handout, we will deal with perimeter problems involving rectangles, squares, and triangles. We will use L and W to denote the length and width, respectively, of a rectangle; we'll use S to represent the length of a side of a square; and we'll use A, B, and C to represent the lengths of the three sides of a triangle. Formulas for the perimeter P of each are shown below. Rectangle Square Triangle W S B C L S A P= 2W + 2L P= 4S P= A+ B+ C Use the following procedure to set up perimeter problems: 1. Read the problem carefully several times. 2. Draw a figure to help visualize the problem. 3. Identify the unknown measures. 4. Represent one of the unknown measures by a suitable letter, such as. Then represent the other unknown measures in terms of. 5. Translate the problem into a linear equation using the appropriate perimeter formula or formulas. 6. Solve the equations 7. State the answer or answers to the problem. 8. Check the answer or answers in the original statement of the problem, not the equation that you set up. Now, let s look at some eample problems involving the perimeter of rectangles, square, and triangles. Revised 03/09 1
2 Eample l: The length of a rectangle is nine inches more than its width. If the perimeter of the rectangle is 62 inches, find its length and width. Let = the width W. Then + 9 = the length L since it is given to be 9 inches more than the width. +9 Since the perimeter P of a rectangle is given by P= 2W + 2L, and the perimeter is 62, 62= 2+ 2( + 9) 62= = = 4 11 The width of the rectangle is 11 inches. + 9= 20 The length of the rectangle is 20 inches. Eample 2: The longest side of a triangle is one foot more than twice the shortest side. The other side is eight feet less than the longest side. If the perimeter of the triangle is 54 feet, find the length of each side. Let the length of the shortest side. Then 2+ 1= the length of the longest side and (2+ 1) 8 or 2 7= the length of the other side. 27 Since the perimeter P of a triangle is given by P= A+ B+ C, and the perimeter is 54, 54 = + (2+ 1) + (2 7) 54= = 5 12 The length of the shortest side is 12 feet. 2+ 1= 25 The length of the longest side is 25 feet. 2 7= 17 The length of the other side is 17 feet. 2+1 Revised 03/09 2
3 Eample 3: A rectangle is formed by placing two identical squares side by side. If the perimeter of the rectangle is 90 centimeters, what is the length of a side of each square? Let = the length of a side of each square. The rectangle that is formed has a width of and a length of 2. Since the perimeter of a rectangle is given by P = 2 W + 2 L and the perimeter is 90, 90= = 6 15 The length of a side of each square is 15 centimeters. Eample 4: In a certain rectangle, the length is three inches more than the width. If the length is increased by four inches and the width is increased by two inches, the perimeter of the new rectangle will be 78 inches. Find the dimensions of the original rectangle. Let = the width of the original rectangle. Then + 3 = the length of the original rectangle. Since the width of the original rectangle is increased by 2, the new width is + 2; and since the length of the original rectangle is increased by 4, the new length is ( + 3) + 4 or + 7. Original Rectangle New Rectangle Using the dimensions of the new rectangle and substituting into the perimeter formula of a rectangle, P= 2W + 2L 78= 2( + 2) + 2( + 7) 78= = The width of the original rectangle is 15 inches. + 3= 18 The length of the original rectangle is 18 inches Revised 03/09 3
4 Eample 5: An 8 by 12 inch picture has a frame of uniform width. If the outer edge of the frame has a perimeter of 52 inches, how wide is the frame? Let = the width of the frame. It can be observed in the figure at the right that a frame of uniform width around the picture increases the dimensions of the 8 by 12 inch picture by 2 inches in each direction. The outer dimensions are (8+ 2 ) by (12+ 2 ) inches. Given the perimeter of the outer edge is 52, we form the equation 2(8+ 2 ) + 2(12+ 2 ) = = The width of the frame is 1.5 inches Eample 6: The length of a side of a square is twice the width of a rectangle, and the length of the rectangle is 21 feet. If the perimeter of the square is the same as the perimeter of the rectangle, find the dimensions of the square and the rectangle. Let = the width of the rectangle. Since the length of a side of the square is twice the width of the rectangle, 2 = the length of a side of the square The perimeter of the square is 4 2 or 8. The perimeter of the rectangle is or Since these perimeters are given to be the same, The dimensions of the rectangle (in feet) are 7 by The length of a side of the square is 14 feet. Revised 03/09 4
5 PERIMETER PROBLEM EXERCISES 1. The perimeter of a rectangle is 52 feet. The length of the rectangle is 10 feet longer than the width. Find the dimensions of the rectangle. 2. The length of a rectangle is three times its width, and its perimeter is 88 inches. Find its length and width. 3. The width of a rectangle is 12 centimeters less than its length, and its perimeter is 48 centimeters. Find the dimensions. 4. The length of a rectangle is four feet more than twice the width. If the perimeter is 110 feet, find the length and the width. 5. The length of a rectangle is three feet less than two times the width. The perimeter is 27 feet. Find the dimensions. 6. One side of a triangle is twice another side. The third side is eight centimeters and the perimeter is 26 centimeters. How long is each side? 7. The longest side of a triangle is five meters more than the second side. The shortest side is two meters less than the second side. If the perimeter of the triangle is 24 meters, find the length of the shortest side. 8. The perimeter of a triangle is 58 meters. The longest side is three times the shortest side. The other side is five meters shorter than the longest side. How long is each side? 9. A rectangle is formed by placing two equal squares side by side. If the perimeter of the rectangle is 30 inches, how long is a side of each square? 10. The perimeter of an equilateral triangle (a triangle whose sides have the same length) is 28 inches more than the length of a side. Find the length of a side. 11. One side of a triangle is si meters less than the second side and twice the length of the third side. If the perimeter is 106 meters, find the length of the three sides. 12. The longest side of a triangle is 12 inches longer than the shortest side. The other side is five inches shorter than the longest side. If the perimeter is 61 inches, find the length of each side. 13. One side of a triangle is three feet longer than another, and the third side is four feet less than the sum of the first two. If the perimeter is 54 feet, find the lengths of the sides. 14. The length of a rectangle is three feet more than twice the width. If the width is increased by four feet, the perimeter of the new rectangle is 68 feet. Find the dimensions of the original rectangle. Revised 03/09 5
6 15. In a certain rectangle, the length is si yards more than the width. If the length is increased by 25 yards and the width decreased by one yard, the perimeter of the new rectangle will be twice that of the original. Find the length and width of the original rectangle. 16. A 5 by 7 inch rectangular picture has a frame of uniform width. If the outer edge of the frame is to have a perimeter of 32 inches, how wide should the frame be? 17. A man plans to build a fence along the four sides of his house. The house is a rectangle that is 40 feet long and 32 feet wide. How far should he uniformly build the fence from the house if he has enough material for 200 feet of fence? 18. A 10 by 12 inch rectangular picture is framed with a frame of uniform width. How wide is the frame if the perimeter of its outer edges is 58 inches? 19. The side of a square is the same length as the shortest side of a triangle. The other two sides of the triangle have equal lengths and are four inches longer than the shortest side of the triangle. Find the lengths of the sides of the triangle and the square if the perimeters of the square and triangle are the same. 20. The length of the longest side of a triangle is ten centimeters more than that of the shortest side. The other side is five centimeters less than the longest side. The perimeter of the triangle is the same as the perimeter of a square with sides the same length as the shortest side of the triangle. Find the lengths of the sides of the triangle and the square. 21. The lengths of the three sides of a triangle are 16, 19, and 24 inches. The shortest side is to be decreased by a certain amount, the net longer side is decreased by twice this amount, and the longest side is decreased by three times this amount to form a triangle with a perimeter of 44 inches. Find the lengths of the sides of the new triangle. 22. A triangle has sides of lengths of 8, 12, and 16 inches. Each side is to be decreased by the same amount to form a new triangle that will have a perimeter onehalf of the original triangle. How much should each side be decreased? 23. The length of a rectangle is four times its width. The perimeter of the rectangle is 84 feet more than the perimeter of a square with a side the same length as the width of the rectangle. Find the dimensions of the rectangle. 24. The length of a side of a square is twice the width of a rectangle. If the perimeter of the square is the same as the perimeter of the rectangle, and the length of the rectangle is 18 feet, find the dimensions of the square and the rectangle. 25. CALCULATOR PROBLEM  The length of a rectangle is 3.46 meters more than the width. If the perimeter is meters, find the dimensions of the rectangle. 26. CALCULATOR PROBLEM  The perimeter of a rectangle is inches, and the length is three times as long as the width. Find the length and width of the rectangle. Revised 03/09 6
7 Solutions to oddnumbered problems and answers to evennumbered problems: 1. Let = the width and + 10= the length 2+ 2( + 10) = = = 18 The width is 8 ft. and the length is 18 ft inches by 33 inches 3. Let = the length and 12= the width 2+ 2( 12) = = = 6 The length is 18 cm and width is 6cm feet by 38 feet 5. Let the width and 2 3= the length 2+ 2(2 3) = = = 8 The width is 5.5 ft and the length is 8 ft. 6. 6, 8, and 12 centimeters 7. Let the length of the 2 nd side, + 5= the length of the longest side, and 2= the length of the shortest side. + ( + 5) + ( 2) = = = 5 The length of the shortest side is 5 meters. 8. 9, 22, and 27 meters Revised 03/09 7
8 9. Let the length of the side of the square. Then the width of the rectangle is and the length is (2 ) = The length of the side of the square is 5 inches inches 11. Let the length of the 3 rd side, 2 the length of the 1 st side and 2+ 6= the length of the 2 nd side (2+ 6) = = = 46 The lengths of the three sides are 20, 40, and 46 meters , 21, and 26 inches 13. Let the length of the 2 nd side, + 3= the length of the 1 st side, and + ( + 3) 4or 2 1= the length of the 3 rd side. + ( + 3) + (2 1) = = = = 25 The lengths of the three sides are 13, 16, and 25 feet by 21 feet 15. Let the width of the original rectangle. so that + 6= the length of the original rectangle. Then + 31 = the length of the new rectangle, and 1= the width of the new rectangle. 2( + 31) + 2( 1) = 2[2+ 2( + 6)] 4+ 60= = = 15 The width is 9 yards and the length is 15 yards. Revised 03/09 8
9 16. 1 inch 17. Let the distance from fence to house. 2(40+ 2 ) + 2(32+ 2 ) = = The distance is 7 feet inches 19. Let the length of the side of the square. Then, also, the length of the shortest side of the triangle. + 4= the length of each of the other two sides of the triangle. 4 + ( + 4) + ( + 4) = 12 The length of a side of the square is 8 inches and the lengths of the three sides of the triangle are 8, 12, and 12 inches. 20. Square: 15 centimeters and Triangle: 15, 20, and 25 centimeters 21. Let = the amount. The sides of the new triangle have lengths 16,19 2 and (16 ) + (19 2 ) + (24 3 ) = = The lengths of the three sides of the new triangle are 13.5, 14, and 16.5 inches inches 23. Let the width. Then 4 the length. 2+ 2(4 ) = The width is 14 ft. and the length is 56 ft. Revised 03/09 9
10 24. Square: 12 feet and Rectangle: 6 by 18 feet 25. Let = the width. Then = the length. 2+ 2( ) = = = 9.03 The width is 5.57 meters and the length is 9.03 meters by inches NOTE: You can get additional instruction and practice by going to the following web sites: This website gives general help in translating word problems into algebraic equations. This website contains many printable worksheets on perimeter and area problems. Revised 03/09 10
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