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Geometry Part 1B Perimeter By Julia Arnold, Dick Gill and Marcia Tharp for Elementary Algebra Math 03 online

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Perimeter is the distance around an object. If the object is a rectangle, then it has 4 sides and opposite sides are equal in length and parallel to each other.The formula for its perimeter is P = 2L + 2W where L is length and W is width of the rectangle. If the object is a circle, we call the perimeter the circumference and ancient mathematicians found its formula to be.

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In many perimeter problems involving rectangles, you may be asked to find the dimensions. The dimensions of a rectangle are the width by length. For example, picture frames are categorized by their dimensions, such as 8 by 10, or 11 by 14. The sum of the dimensions represents half of the perimeter. A picture frame that is an 8 by 10 would have a perimeter of 2(8 + 10) or 36 inches. What would be the perimeter of an 11 by 14 picture frame? The dimensions are only half the perimeter.

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Answer: An 11 by 14 inch picture frame would have a perimeter of 2(11+14) = 50 inches.

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Example 1: Suppose we have a rectangle with a perimeter of 26 inches.We are told that the length is 10 more than twice the width.Can we find the dimensions of the rectangle? Define the variables: Let x = width, then 10 + 2x = length The sum of width and length is half the perimeter, thus... x + 10 + 2x = 26/2 3x + 10 = 13 3x = 3 x = 1 inch This is the width. 2x + 10 = 2(1) + 10 = 12 which is the length. The dimensions are 1 by 12.

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Example 2. The perimeter of a rectangular lot is 440 meters. If the length is 150 m. find the width. Let x = width 2L + 2W = P 2(150) + 2x = 440 300 + 2x = 440 300 + 2x - 300 = 440 - 300 2x = 140 x = 70 m, the width or L + W = P/2 150 + x = 220 x = 220 - 150= 70 m, the width

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Example 3. The length of a rectangle is 10 more than twice the width. Find the length and the width if the perimeter is 56 inches. Go ahead and guess now. Remember that 2L + 2W = P or 2(L+W) = P or L + W = P/2 Let x = the width 2x + 10 = the length 2L + 2W = P 2(2x + 10) + 2x = 56 4x + 20 + 2x = 56 6x + 20 - 20 = 56 - 20 6x = 36 x = 6 inches, the width 2x + 10 = 2(6) + 10 = 22 inches, the length Or 2x + 10 + x = 56/2 3x + 10 = 28 3x = 18 x = 6 inches, the width 2(6)+10=22 inches, the length

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Example 4. The perimeter of a triangle is 85 cm. If the middle side is 5 cm. larger than the smallest side and the large side is twice the smallest side, find each side. 2L + 2W = P only works for rectangles. Perimeter is the distance around the outside of the figure. For a triangle then, the perimeter is the sum of the three sides. Let x = the smallest side x + 5 = the middle side 2x = the largest side x + (x + 5) + 2x = 85 4x + 5 - 5 = 85 - 5 4x = 80 x = 20 cm, the shortest side x + 5 = 25 cm, the middle side and 2x = 40 cm

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How would you find the perimeter of this architect’s home plan?

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10’ This shape consists of a rectangle with two Overlapping circles. Can we find the perimeter If we know the dimensions of the rectangle? 20’ Example 5:

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This shows the outside of the figure which Is the perimeter that we must find. The perimeter Is actually two lengths of the rectangle and two Semi-circles. We need the formula for the circumference of a circle which is C = 2 pi( r) Where r is the radius of the circle. An alternate formula is C = pi d where d is the diameter of the circle. pi is the number which is approximated by 3.141592654 20’ 10’

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Since the two semicircles are the same size the two parts make one complete circle. Thus the perimeter is P = 20 + 20 + (10) = 40 + 10 Which is approximately 71.4’ 20’ 10’ Is the Greek symbol for Pi

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Other four sided figures which have opposite sides equal are: parallelograms which have opposite sides parallel as well. A rhombus which has four equal sides with opposite sides parallel

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What is the difference between rectangles, squares, parallelograms and rhombuses? A square is a rectangle which has 4 equal sides and every angle is a right angle. Square Rectangle

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A rectangle is a parallelogram because opposite sides are equal in both and opposite sides are parallel in both. The difference between the two is the rectangle has four right angles. A parallelogram has no right angles.

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A square is a rhombus, because all four sides are equal and opposite sides are parallel A rhombus is also a parallelogram because opposite sides are equal and parallel. A square has four right angles. A rhombus has no right angles.

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If a figure is called a quadrilateral, then that means it has four sides, but they could all be different measurements and they do not have to have opposite sides parallel.

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Your Turn Work out the problems on the next slides to see if you understand this concept.

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1. The perimeter of a rectangular lot is 440 meters. If the length is 150 m. find the width. Complete Solution 2. The perimeter of a triangle is 85 cm. If the middle side is 5 cm. larger than the smallest side and the large side is twice the smallest side, find each side. Complete Solution Complete Solution 3. A rectangular lot has a perimeter of 80 feet and length of 25 feet. What is the width of the lot? 4. Mr. Green wants to make a rectangular garden plot. He wants the length to be twice the width plus 3 feet. If the perimeter of the lot is 66 feet, what are the dimensions of the lot? Complete Solution 5. Mr. Fixit wants to fence in a rectangular lot where one side will be the wall of his barn. He has 60 feet of fencing available. If the length is 4 feet less than twice the width. What are the dimensions of the lot? Complete Solution

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6. A quadrilateral has sides measuring 16.5 in., 17.3 in., 21.8 in., 29.2 in. What is its perimeter? 7. An Equilateral triangle has a side measuring 1.28 cm. What is its perimeter? 8. An Isosceles triangle has its equal side measuring 15.4 in. and its 3rd side measuring 23.9 in. What is its perimeter? 9. A square measures 1.63 m. What is its perimeter? 10. A rhombus measures 8.06 ft. on a side. What is its perimeter? 11. A parallelogram measures 47.2 on one side and 36.8 on another. What is its perimeter? 12. A rectangle has dimensions 57.9 by 39.8 in. What is its perimeter? Complete Solutions When complete go to next slideWhen complete go to next slide.

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13. Find the circumference of the circle whose radius is 14.3 cm. Round to the nearest tenth. 14. Find the circumference of the circle whose diameter is 8.4 in. Round to the nearest tenth. Find the perimeters of the indicated figures: 15. 12 ft. 6 ft. 7 ft. 11 ft. 18 ft. 16. 60.8 ft. 46.0 ft. 17. 28 cm. 12 cm. 14 cm. 16 cm 12 cm Complete Solutions

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18.15.3 cm. 19.6 cm Complete Solutions Find the perimeter of this figure.

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1. The perimeter of a rectangular lot is 440 meters. If the length is 150 m. find the width. 2. The perimeter of a triangle is 85 cm. If the middle side is 5 cm. larger than the smallest side and the large side is twice the smallest side, find each side. Length + width = 1/2 perimeter 150 + x = 440/2 150 + x = 220 x = 70 The width is 70 X = smallest side x + 5 = middle side 2x = largest side x + x + 5 + 2x = 85 4x + 5 = 85 4x = 80 x = 20 X = 20 x + 5 = 25 2x = 40 check 20 + 25 + 40 = 85 Return to Problems

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Let x = width x + 25 = 80/2 x + 25 = 40 x = 15 The width of the lot is 15 feet. Let x = width, length = 2x + 3 x + 2x + 3 = 66/2 3x + 3 = 33 3x = 30 x = 10 2x + 3 = 23 The dimensions are 10 x 23 feet. Return to Problems 4. Mr. Green wants to make a rectangular garden plot. He wants the length to be twice the width plus 3 feet. If the perimeter of the lot is 66 feet, what are the dimensions of the lot?

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Barn Wall Three sides fencing Let x = width 2x – 4 = length 2x-4 x x Add 3 sides: x + x + 2x – 4 = 60 4x – 4 = 60 4x = 64 x = 16 2x – 4 = 28 The dimensions are 16 by 28 feet. Return to Problems

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. 6. A quadrilateral has sides measuring 16.5 in., 17.3 in., 21.8 in., 29.2 in. What is its perimeter? 84.8 inches 7. An Equilateral triangle has a side measuring 1.28 cm. What is its perimeter? 1.28*3= 3.84 cm 8. An Isosceles triangle has its equal side measuring 15.4 in. and its 3rd side measuring 23.9 in. What is its perimeter? 15.4*2+23.9 = 54.7 in. 9. A square measures 1.63 m. What is its perimeter? 1.63*4 = 6.52 m 10. A rhombus measures 8.06 ft. on a side. What is its perimeter? 8.06*4 = 32.24 ft. 11. A parallelogram measures 47.2 on one side and 36.8 on another. What is its perimeter? 47.2*2+36.8*2 = 168 12. A rectangle has dimensions 57.9 by 39.8 in. What is its perimeter? 2(57.9+39.8)= 195.4 Return to Problems

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13. Find the circumference of the circle whose radius is 14.3 cm. Round to the nearest tenth. Ans. 2 (14.3) = 89.8 14. Find the circumference of the circle whose diameter is 8.4 in. Round to the nearest tenth. Ans (8.4) =26.4 Find the perimeters of the indicated figures: 15. 12 ft. 6 ft. 7 ft. 11 ft. 18 ft. 16. 60.8 ft. 46.0 ft. 17. 28 cm. 12 cm. 14 cm. 16 cm 12 cm 6+7+11+18+12= 54 ft. 60.8*2+46+46*pi/2=239.9 ft. 3/4 of a rectangle plus 1/2 of a circle 28+14+16= 58 58 + 12+ 28+12+14+ 12+16+12= 164cm Return to Next Problems Return to same questions

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18.15.3 cm. 19.6 cm This shape is a combination of a rectangle and two semi-circles. The two semi-circles = one circle. Top of Rectangle + bottom of rectangle + circumference of circle = perimeter 15.3 + 15.3 + 19.6 = 30.6 + 19.6 = 92.2 cm Back to beginning of questions End show

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