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**Respect Responsibility
Readiness**

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

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 cm^{2}. 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 cm^{2}.
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 m^{2}. 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 cm^{2 }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 cm^{2}. Find the
dimensions of the rectangle.

20. The length of a
rectangle is 4 cm more than the width. The area is 96 cm^{2}. Find the
dimensions of the rectangle.

21. The length of a
photograph is 1 cm less than twice the width. The area is 45 cm^{2}.
Find the dimensions of the photograph.

22. If the sides of a
square are increased by 3 m, the area becomes 64 m^{2}. 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 m^{2}. 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 m^{2}. 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 m^{2}.
Find the length and width.

28. The length of a
rectangle is three times the width. The area is 108 cm^{2}. Find the
dimensions of the rectangle.

29. The length of a
photograph is 1 cm less than twice the width. The area is 28 cm^{2}.
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
m^{2}. 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 m^{2}.
Find the dimensions of the mural.

32. The length of a
rectangle is 6 cm more than the width. The area is 11 cm^{2}. Find the
length and width.

33. The length of a
rectangular garden is 4 m greater than the width. The area is 71 m^{2}.
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 km^{2}.
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 cm^{2}.
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 cm^{2}. 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: a^{2 }+ b^{2 =
}c^{2}.) 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 m^{2}, 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.