CHECK IF THE GIVEN FIGURES ARE SIMILAR

Are the two figures similar ? If so, state the scale factor.

Problem 1 :

Solution :

Comparing the corresponding sides, we get

Since the ratios are equal, the given shapes are similar. The scale factor is 3 :10.

Problem 2 :

Solution :

Comparing the corresponding sides,

40/8 = 5

50/20 ≠ 5

Since the ratios are not same, the shapes are not similar.

Problem 3 :

Solution :

Since the ratios are same, the shapes are similar. The required scale factor is 1 : 3.

Problem 4 :

Solution :

Comparing the corresponding sides,

18/54 = 1/3

6/21 = 2/7

Since the ratios are not same, they are not similar shapes.

Each pair of figures is similar. Use the information given to find the scale factor of the figure on the left to the figure on the right. 

Problem 5 :

Solution :

Surface area of large shape : Surface area of small shape

= 36 : 25

k² = 36/25

k = √36/25

k = 6/5

So, the scale factor is 6 : 5.

Problem 6 :

Solution :

Surface area of small shape : Surface area of large shape

= 7π : 175π

k² = 7π/175π

k² = 1/25

k = √1/25

k = 1/5

So, the scale factor is 1 : 5.

Problem 7 :

Solution :

Volume of small shape : Volume of large shape

10240 : 20000

k³ = 10240/20000

k³ = 64/125

k = ∛(64/125)

k = 4/5

So, the scale factor is 4 : 5.

Problem 8 :

Solution :

Volume of large shape : Volume of small shape

= 3240 : 120

k³ = 3240/120

k³ = 27

k = ∛27

k = 3

So, the scale factor is 3 : 1.

Problem 9 :

A person that is 6 feet tall casts a 3-foot-long shadow. A nearby palm tree casts a 15-foot-long shadow. What is the height h of the palm tree? Assume the triangles are similar.

Solution :

The above triangles are similar, 

h/15 = 6/3

h/15 = 2

h = 2(15)

h = 30 ft

So, the height of tree is 30 ft.

Problem 10 :

A swimming pool is similar in shape to a volley ball court. What is the area A of the pool ?

Solution :

When shapes are similar, the square of the ratio of similar sides is equal to the ratio of the areas.

(18/10)² = Area pool / area of volley ball court

(18/10)² = Area pool / 200

324/100 = Area pool / 200

Area of pool = 3.24(200)

= 648

So, area of pool is 648 square yards.

Problem 11 :

The ratio of the perimeters is 7:10

Solution :

When two shapes are similar, the ratio of the corresponding sides will be equal to ratio of their perimeters.

x : 12 = 7 : 10

x/12 = 7/10

Doing cross multiplication, we get

10x = 7(12)

x = 84/10

x = 8.4

Problem 12 :

The ratio of the perimeters is 8 : 5.

Solution :

x : 16 = 8 : 5

x/16 = 8/5

x = (8/5) ⋅ (16)

x = 25.5

Problem 13 :

What is the height x of the flagpole? Assume the triangles are similar.

Solution :

Height of the flag pole is x ft.

x / 28 = 5 / 4

x = (5/4) ⋅ 28

= 35

So, the height of the flag pole is 35 ft.

Problem 14 :

A rectangular school banner has a length of 44 inches and a perimeter of 156 inches. The cheerleaders make signs similar to the banner. The length of a sign is 11 inches. What is its perimeter?

Solution :

Ratio of corresponding sides = 44 : 11

Perimeter of the school banner = 156 inches

Perimeter of the sign. Let P be the required perimeter.

156 : P = (44 : 11)²

156/P = (44/11)²

156/P = (16/1)

156 = 16 P

P = 156/16

P = 9.75

So, the required perimeter is 9.75 inches.

Problem 15 :

The ratio of the side length of Square A to the side length of Square B is 4 : 9. The side length of Square A is 12 yards. What is the perimeter of Square B?

Solution :

Side length of square A : Side length of square B = 4 : 9

Side length of square A = 12 yards

Let x be the side length of square B.

12 : x = 4 : 9

12/x = 4/9

4x = 12(9)

x = 108/4

x = 27

Side length of square B = 27 yards

Perimeter of square B = 4(27)

= 108 yards.