Perimeter and Area

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1. The length of a rectangle is 3 times the width. The perimeter is 96 cm.

Solution

First draw a picture.

Picture coming

Let w = width and 3w = length

Perimeter = 2 lengths + 2 widths or P = 2l + 2w

P = 2l + 2w

l = 3w

Substitute 3w for l

2(3w) + 2w = 96

6w + 2w = 96

8w = 96

w = 12

So the width is 12 and the length is 36 

Check

36 + 12 + 36 + 12 = 96

96 = 96

 

 

2. The width of a rectangle is 12 cm less than the length. The perimeter is 156 cm. Find the width and length.

Solution

First draw a picture.

Picture coming

Let l = length and l - 12 = width

Perimeter = 2 lengths + 2 widths or P = 2l + 2w

P = 2l + 2w

w = l - 12

Substitute l – 12 for w

P = 2l + 2(l – 12)

156 = 2l + 2l – 24

156 = 4l – 24

180 = 4l

l = 45

So the length is 45 and the width is 33 

Check

P = 2l + 2w

156 = 2(45) + 2(33)

156 = 90 + 66

156 = 156

 

 

3. The length of a rectangle is 2 cm less than 7 times the width. The perimeter is 60 cm. Find the width and length.

Solution

First draw a picture.

Picture coming

Let w = width and (7w – 2) = length

Perimeter = 2 lengths + 2 widths or P = 2l + 2w

P = 2l + 2w

l = 7w – 2

Substitute (7w – 2) for l

P = 2(7w – 2) + 2w

60 = 14w – 4 + 2w

60 = 16w – 4

64 = 16w

w = 4

So the width is 4 and the length is 26 

Check

26 + 4 + 26 + 4 = 60

60 = 60

 

 

4. The perimeter of a triangle is 76 cm. Side a of the triangle is twice as long as side b. Side c is 1 cm longer than side a. Find the length of each side.

Solution

First draw a picture.

Picture coming

Let a, b and c represents the 3 sides

a = 2b or b = (1/2)a

c = a + 1

Perimeter = a + b + c

P = a + b + c

Substitute in for b and c

P = a + (1/2)a + (a + 1)

76 = 2.5a + 1

75 = 2.5a

a = 30

So a = 30, b = 15, c = 31 

Check

30 + 15 + 31 = 76

76 = 76

 

 

5. The first side of a triangle is 8 m shorter than the second side. The third side is 4 times as long as the first side. The perimeter is 26 m. Find the length of each side.

Solution

First draw a picture.

Picture coming

Let a, b and c represents the 3 sides

Let a = first side

Let b = second side

Let c = third side

b = a + 8

c = 4a

Perimeter = a + b + c

P = a + b + c

Substitute in for b and c

P = a + (a + 8) + (4a)

26 = 6a + 8

18 = 6a

a = 3

So a = 3, b = 11, c = 12 

Check

3 + 11 + 12 = 26

26 = 26

 

 

6. A triangular sail has a perimeter of 25 m. Side a is 2 m shorter than twice side b, and side c is 3 m longer than side b. Find the length of each side..

Solution

First draw a picture.

Picture coming

Let a, b and c represent the 3 sides

a = 2b - 2

c = b + 3

Perimeter = a + b + c

P = a + b + c

Substitute in for a and c

P = (2b – 2) + b + (b + 3)

25 = 4b + 1

24 = 4b

b = 6

So b = 6, a = 10, c = 9

Check

10 + 6 + 9 = 25

25 = 25

 

7. The triangle shown at the right is isosceles. That is, it has two sides of equal length. The third side is 30 m shorter than twice the length of each congruent side. The perimeter is 570 m. Find the length of each side.

Solution

First draw a picture.

Picture coming

Let a, a and b represent the 3 sides

a represents the 2 equal sides

b = 2a -30

Perimeter = a + a + b

P = a + a + b

Substitute in for b

570 = a + a + (2a -30)

570 = 4a - 30

600 = 4a

a = 150

So a = 150 and b = 270

Check

150 + 150 + 270 = 570

570 = 570

 

8. The length of a rectangle is 3 times the width. If the length is decreased by 4 m and the width is increased by 1 m, the perimeter will be 66 m. Find the dimensions of the original rectangle.

9. The length of a rectangle is 6 cm longer than the width. If the length is increased by 9 cm and the width by 5cm, the perimeter will be 160 cm. Find the dimensions of the original rectangle.

10. The length of a rectangle is 7 m less than twice the width. If the length is decreased by 1 m and the width by 4 m, the perimeter will be 66 m. Find the dimensions of the original rectangle.

11. The perimeter of a triangle is 69 cm. Side a is 5 cm shorter than side b. Side c is twice as long as side a. Find the length of each side.

12. The first side of a triangle is 7 cm shorter than twice the second side. The third side is 4 cm longer than the first side. The perimeter is 80 cm. Find the length of each side.

13. The length of a rectangular field is 18 m longer than the width. The field is enclosed with fencing and divided into two parts with a fence parallel to the shorter sides. If 216 m of fencing are required, what are the dimensions of the outside rectangle? (See diagram to the right.)

14. The length of a rectangle is 3 cm greater than the width. If each dimension is increased by 2 cm, the area is increased by 26 cm2. Find the original dimensions of the rectangle.

15. The length of a rectangle is 2 cm greater than the width. If the width is increased by 3 cm, and the length is increased by 4 cm, the area is increased by 88 cm2. Find the original dimensions of the rectangle.

16. A rectangular garden is 4 m longer than it is wide. If the width is decreased by 1 m, and the length is increased by 5 m, the area is increased by 15 m2. Find the original dimensions of the garden.

17. A rectangular swimming pool is 2 m longer than it is wide. If the width is decreased by 3 m, and the length is increased by 4 m, the area remains the same as the original area, Find the original dimensions of the pool. 

18. A rectangular picture is 6 cm longer than it is wide. A frame 1 cm wide is placed around the picture. The area covered by the picture and frame together is 48 cm2 greater than the area of the picture alone. Find the dimensions of the picture.

19. The length of a rectangle is 3 cm more than the width. The area is 70 cm2. Find the dimensions of the rectangle.

20. The length of a rectangle is 4 cm more than the width. The area is 96 cm2. Find the dimensions of the rectangle.

21. The length of a photograph is 1 cm less than twice the width. The area is 45 cm2. Find the dimensions of the photograph.

22. If the sides of a square are increased by 3 m, the area becomes 64 m2. Find the length of a side of the original square.

23. A square field had 5 m added to its length and 2 m added to its width. The field then had an area of 130 m2. Find the length of a side of the original held. 

24. The dimensions of a rectangular garden were 4 m by 5 m. Each dimension was increased by the same amount. The garden then had an area of 56 m2. Find the dimensions of the new garden. (Hint: Let x be the amount of increase.)

25. The dimensions of a rectangular garden were 3 m by 10 m. When both dimensions were increased by equal amounts, the area of the garden doubled. Find the dimensions of the new garden.

26. A 4 m by 6 m rug covers half of the floor area of a room and leaves a uniform strip of bare floor around the edges. What are the dimensions of the room?

27. The length of a rectangle is 4 m more than the width. The area of the rectangle is 45 m2. Find the length and width.

28. The length of a rectangle is three times the width. The area is 108 cm2. Find the dimensions of the rectangle.

29. The length of a photograph is 1 cm less than twice the width. The area is 28 cm2. Find the dimensions of the photograph.

30. A square field had 3 m added to its length and 2 m added to its width. The field then had an area of 90 m2. Find the length of a side of the original field.

31. The length of a rectangular mural is 2 m greater than the width. The area is 20 m2. Find the dimensions of the mural.

32. The length of a rectangle is 6 cm more than the width. The area is 11 cm2. Find the length and width.

33. The length of a rectangular garden is 4 m greater than the width. The area is 71 m2. Find the dimensions of the garden.

34. The length of a rectangular park is 2 km less than twice the width. The area is 9 km2. Find the dimensions of the park.

35. The base of a triangle is 3 cm longer than its altitude. The area of the triangle is 35 cm2. Find the altitude. (Hint: The area of a triangle equals ˝ base•altitude.)

36. The altitude of a triangle is 2 cm shorter than its base. The area is 15 cm2. Find the base of the triangle.

37. A flower garden is in the shape of a right triangle. The longest side of the triangle measures 13 m. One of the shorter sides is 7 m longer than the other. Find the length of the shortest side. (Hint: Use the Pythagorean Theorem: a2 + b2 = c2.) The diagonal measure of a movie screen is 6 m. The length of the screen is 2 m greater than the height. Find the dimensions of the screen.

38. A square picture is mounted in a frame 1 cm wide. The area of the picture is of the total area. Find the length of a side of the picture.

39. A rectangular pond measures 3 m by 5 m. A concrete walk of uniform width is constructed around the pond. If the walk and pond together cover an area of 39 m2, how wide is the walk?

40. A rectangular counter is covered with 600 square tiles. The counter could have been covered with 400 tiles 1 cm longer on a side. Find the length of a side of the smaller tile.

41. Find the width and length. The length of a rectangle is 5 m greater than the width. The perimeter is 150 m. Find the width and length.