Perimeter and Area

- 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

- 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

- 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

- 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

- 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

- 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

- The triangle shown at the right is
. 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.*isosceles*

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

- 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.
- 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.
- 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.
- The perimeter of a triangle is 69 cm. Side
is 5 cm shorter than side*a*. Side c is twice as long as side*b*. Find the length of each side.*a* - 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.
- 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.)
- 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. - 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. - 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. - 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.
- 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. - The length of a rectangle is 3 cm more than the width. The area is 70 cm
^{2}. Find the dimensions of the rectangle. - The length of a rectangle is 4 cm more than the width. The area is 96 cm
^{2}. Find the dimensions of the rectangle. - 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. - 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. - 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. - 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.) - 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.
- 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?
- 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. - The length of a rectangle is three times the width. The area is 108 cm
^{2}. Find the dimensions of the rectangle. - 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. - 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. - 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. - The length of a rectangle is 6 cm more than the width. The area is 11 cm
^{2}. Find the length and width. - 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. - 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. - 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 ½ base•altitude.) - The altitude of a triangle is 2 cm shorter than its base. The area is 15 cm
^{2}. Find the base of the triangle. - 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. - 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.
- 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? - 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.
- 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.