# Study Island

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1. Clarise drew the scale drawing below of a circular fish pond she plans to put in her backyard.

[pic]

If 1 centimeter on the scale drawing represents 2 feet, what will be the diameter of Clarise's fish pond?

|[pic|A. |6 feet |

|] | | |

|[pic|B. |12 feet |

|] | | |

|[pic|C. |37.68 feet |

|] | | |

|[pic|D. |56.52 feet |

|] | | |

[pic]

2.

[pic]

Picture is not drawn to scale.

The dimensions of triangle A are twice the dimensions of triangle B. The area of triangle A is 72 cm2.

What is the area of triangle B?

|[pic|A. |18 cm2 |

|] | | |

|[pic|B. |72 cm2 |

|] | | |

|[pic|C. |216 cm2 |

|] | | |

|[pic|D. |36 cm2 |

|] | | |

[pic]

3. Delisa is designing a website that will be viewable on both computers and mobile devices, so the website as it is seen on a mobile device is proportional to the website as it is seen on a computer. The diagonal of Delisa's computer monitor is 21 inches, and the diagonal of Delisa's tablet is 9.6 inches.

If an image is 17 centimeters wide on the website when it's displayed on Delisa's computer, how wide should the image be on the tablet? Round to the nearest tenth of a millimeter, if necessary.

|[pic|A. |1.3 millimeters |

|] | | |

|[pic|B. |77.7 millimeters |

|] | | |

|[pic|C. |1,422 millimeters |

|] | | |

|[pic|D. |371.9 millimeters |

|] | | |

[pic]

4.

[pic]

Above are two different models of the same triangular handkerchief. If the area of the model on the right is 108 sq cm, what is the area of the model on the left?

|[pic|A. |12 sq cm |

|] | | |

|[pic|B. |27 sq cm |

|] | | |

|[pic|C. |54 sq cm |

|] | | |

|[pic|D. |36 sq cm |

|] | | |

[pic]

5. The following blueprint of a kitchen has dimensions of 7 inches by 7 inches. The island has been highlighted in red.

[pic]

The island's actual dimensions are [pic]feet by [pic]feet. If the scale of the blueprint is 1 inch = [pic]feet, what are the dimensions of the island on the blueprint?

|[pic|A. |[pic] |

|] | | |

|[pic|B. |[pic] |

|] | | |

|[pic|C. |[pic] |

|] | | |

|[pic|D. |[pic] |

|] | | |

[pic]

6. Figure STUV and figure WXYZ, shown below, are similar figures.

[pic]

The scale factor of figure STUV to figure WXYZ is 3:1. If ST = 78 mm and SV = 102 mm, what is the length of side WZ?

|[pic|A. |306 mm |

|] | | |

|[pic|B. |234 mm |

|] | | |

|[pic|C. |26 mm |

|] | | |

|[pic|D. |34 mm |

|] | | |

[pic]

7.

[pic]

Above are two different models of the same triangle. If the area of the model on the right is 9 sq in, what is the area of the model on the left?

|[pic|A. |4.5 sq in |

|] | | |

|[pic|B. |45 sq in |

|] | | |

|[pic|C. |36 sq in |

|] | | |

|[pic|D. |18 sq in |

|] | | |

[pic]

8.

[pic]

Above are two different models of the same triangle. If the area of the model on the left is 21 sq cm, what is the area of the model on the right?

|[pic|A. |42 sq cm |

|] | | |

|[pic|B. |23.5 sq cm |

|] | | |

|[pic|C. |105 sq cm |

|] | | |

|[pic|D. |84 sq cm |

|] | | |

[pic]

9.

|[pic] |[pic] |

|[pic] | |

| |[pic] |

Above are two different models of a microchip. If the model of the microchip on the left measures 6 cm tall, how tall is the model of the microchip on the right?

|[pic|A. |10 cm |

|] | | |

|[pic|B. |16 cm |

|] | | |

|[pic|C. |12 cm |

|] | | |

|[pic|D. |14 cm |

|] | | |

[pic]

10.

[pic]

Picture is not drawn to scale.

The dimensions of rectangle A are four times the dimensions of rectangle B. The area of rectangle B is 18 cm2.

What is the area of rectangle A?

|[pic|A. |288 cm2 |

|] | | |

|[pic|B. |144 cm2 |

|] | | |

|[pic|C. |9 cm2 |

|] | | |

|[pic|D. |72 cm2 |

|] | | |

[pic]

11.

[pic]

Picture is not drawn to scale.

The dimensions of triangle A are three times the dimensions of triangle B. The area of triangle A is 90 cm2.

What is the area of triangle B?

|[pic|A. |270 cm2 |

|] | | |

|[pic|B. |30 cm2 |

|] | | |

|[pic|C. |15 cm2 |

|] | | |

|[pic|D. |10 cm2 |

|] | | |

[pic]

12.

[pic]

Picture is not drawn to scale.

The dimensions of rectangle B are twice the dimensions of rectangle A. The area of rectangle A is 98 cm2.

What is the area of rectangle B?

|[pic|A. |294 cm2 |

|] | | |

|[pic|B. |196 cm2 |

|] | | |

|[pic|C. |49 cm2 |

|] | | |

|[pic|D. |392 cm2 |

|] | | |

[pic]

13.

[pic]

Above are two different models of the same hexagon. If the side length of the model on the left is [pic]in, what is the corresponding side length of the model on the right?

|[pic|A. |[pic] |

|] | | |

|[pic|B. |[pic] |

|] | | |

|[pic|C. |[pic] |

|] | | |

|[pic|D. |[pic] |

|] | | |

[pic]

14. Pierce works at a tutoring center on the weekends. At the center, they have a large calculator to use for demonstration purposes that is a scale model of calculators available for the students to use.

Each key on the student calculators is 13 millimeters wide, and each key on the demonstration calculator is 2.6 centimeters wide. If the student calculators are 245 millimeters tall, how tall is the demonstration calculator?

|[pic|A. |49 meters |

|] | | |

|[pic|B. |0.026 meter |

|] | | |

|[pic|C. |6.37 meters |

|] | | |

|[pic|D. |0.49 meter |

|] | | |

[pic]

15.

[pic]

Above are two different models of the same pentagon. If the length of side b is 10 cm, what is the length of side a?

|[pic|A. |12.5 cm |

|] | | |

|[pic|B. |5 cm |

|] | | |

|[pic|C. |6.25 cm |

|] | | |

|[pic|D. |2.5 cm |

|] | | |

[pic]

16.

[pic]

Above are two different models of the same rectangle. If the area of the model on the left is 72 sq cm, what is the area of the model on the right?

|[pic|A. |8 sq cm |

|] | | |

|[pic|B. |24 sq cm |

|] | | |

|[pic|C. |216 sq cm |

|] | | |

|[pic|D. |66 sq cm |

|] | | |

[pic]

17. Trisha drew a blueprint of her bedroom, shown below. Denoted by a 5 on the blueprint, her nightstand's blueprint dimensions are [pic]inch by [pic]inch.

[pic]

If the scale of the blueprint is 1 inch = 2 feet, what are the actual dimensions of Trisha's nightstand?

|[pic|A. |[pic] |

|] | | |

|[pic|B. |[pic] |

|] | | |

|[pic|C. |[pic] |

|] | | |

|[pic|D. |[pic] |

|] | | |

[pic]

18.

[pic]

Picture is not drawn to scale.

The dimensions of triangle B are three times the dimensions of triangle A. The area of triangle A is 12.5 cm2.

What is the area of triangle B?

|[pic|A. |37.5 cm2 |

|] | | |

|[pic|B. |112.5 cm2 |

|] | | |

|[pic|C. |56.25 cm2 |

|] | | |

|[pic|D. |25 cm2 |

|] | | |

[pic]

19.

[pic]

Picture is not drawn to scale.

The dimensions of circle B are twice the dimensions of circle A. The area of circle B is 324[pic] cm2.

What is the area of circle A?

|[pic|A. |162[pic] cm2 |

|] | | |

|[pic|B. |324[pic] cm2 |

|] | | |

|[pic|C. |81[pic] cm2 |

|] | | |

|[pic|D. |40.5[pic] cm2 |

|] | | |

[pic]

20. Preston drew a blueprint of an additional room he is planning to add to his house. On the blueprint, 1 centimeter is equal to 3 feet in the actual room. If Preston wants the room to measure 9 feet by 12 feet, what are these dimensions on the blueprint?

|[pic|A. |3 cm by 8 cm |

|] | | |

|[pic|B. |4 cm by 6 cm |

|] | | |

|[pic|C. |3 cm by 4 cm |

|] | | |

|[pic|D. |6 cm by 8 cm |

|] | | |

[pic]

21.

[pic]

Above are two different models of the same television. If the screen in the model on the left has a 14-inch diagonal, what is the diagonal of the screen in the model on the right?

|[pic|A. |56 in |

|] | | |

|[pic|B. |84 in |

|] | | |

|[pic|C. |28 in |

|] | | |

|[pic|D. |42 in |

|] | | |

[pic]

22.

[pic]

Above are two different models of the same rectangular patio. If the area of the model on the left is 2 sq cm, what is the area of the model on the right?

|[pic|A. |10 sq cm |

|] | | |

|[pic|B. |27 sq cm |

|] | | |

|[pic|C. |20 sq cm |

|] | | |

|[pic|D. |50 sq cm |

|] | | |

[pic]

23. The image on a movie poster was shrunk to make the DVD cover art for the movie, so that the cover art is a scale image of the poster. The poster is 48 inches wide, and the DVD cover art is 4 inches wide. If the diagonal of the poster is 8 feet, what is the diagonal of the DVD cover art?

|[pic|A. |0.67 inch |

|] | | |

|[pic|B. |8 inches |

|] | | |

|[pic|C. |16 inches |

|] | | |

|[pic|D. |7 inches |

|] | | |

[pic]

24.

[pic]

Picture is not drawn to scale.

The dimensions of square A are three times the dimensions of square B. The area of square A is 1,764 cm2.

What is the area of square B?

|[pic|A. |588 cm2 |

|] | | |

|[pic|B. |196 cm2 |

|] | | |

|[pic|C. |49 cm2 |

|] | | |

|[pic|D. |392 cm2 |

|] | | |

[pic]

25.

[pic]

Above are two different models of the same triangular-shaped garden. If the height of the model on the left is 17 cm, what is the height of the model on the right?

|[pic|A. |22 cm |

|] | | |

|[pic|B. |19 cm |

|] | | |

|[pic|C. |34 cm |

|] | | |

|[pic|D. |24.5 cm |

|] | | |

[pic]

26.

[pic]

Above are two different models of the same diamond-shaped platform. If the length of side a is 4 cm, what is the length of side b?

|[pic|A. |5.25 cm |

|] | | |

|[pic|B. |10 cm |

|] | | |

|[pic|C. |6 cm |

|] | | |

|[pic|D. |5 cm |

|] | | |

[pic]

27. Figure KLMN and figure PQRS, shown below, are similar figures.

[pic]

If KN = 6 cm, MN = 14 cm, RS = 28 cm, and PS = 12 cm, what is the scale factor of figure KLMN to figure PQRS?

|[pic|A. |[p|

|] | |ic|

| | |] |

|[pic|B. |[p|

|] | |ic|

| | |] |

|[pic|C. |[p|

|] | |ic|

| | |] |

|[pic|D. |[p|

|] | |ic|

| | |] |

[pic]

28.

[pic]

Picture is not drawn to scale.

The dimensions of triangle B are twice the dimensions of triangle A. The area of triangle B is 60 cm2.

What is the area of triangle A?

|[pic|A. |30 cm2 |

|] | | |

|[pic|B. |15 cm2 |

|] | | |

|[pic|C. |67.5 cm2 |

|] | | |

|[pic|D. |45 cm2 |

|] | | |

[pic]

29.

[pic]

Above are two different models of the same rectangle. If the width of the model on the left is 8.25 in, what is the width of the model on the right?

|[pic|A. |16.5 in |

|] | | |

|[pic|B. |2.75 in |

|] | | |

|[pic|C. |3 in |

|] | | |

|[pic|D. |11.25 in |

|] | | |

[pic]

30. The dimensions of an office conference room are 15 feet by 27 feet. If the conference room blueprint dimensions are 5 centimeters by 9 centimeters, what is the scale of the blueprint?

|[pic|A. |1 centimeter = 9 feet |

|] | | |

|[pic|B. |1 centimeter = 3 feet |

|] | | |

|[pic|C. |1 centimeter = 15 feet |

|] | | |

|[pic|D. |1 centimeter = 5 feet |

|] | | |

[pic]

31.

[pic]

Picture is not drawn to scale.

The dimensions of rectangle A are three times the dimensions of rectangle B. The area of rectangle B is 72 cm2.

What is the area of rectangle A?

|[pic|A. |216 cm2 |

|] | | |

|[pic|B. |324 cm2 |

|] | | |

|[pic|C. |36 cm2 |

|] | | |

|[pic|D. |648 cm2 |

|] | | |

[pic]

32.

[pic]

Above are two different diagrams of the same backyard. If the length of the diagram on the left is 6 in, what is the length of the diagram on the right?

|[pic|A. |19 in |

|] | | |

|[pic|B. |2.00 in |

|] | | |

|[pic|C. |18 in |

|] | | |

|[pic|D. |9 in |

|] | | |

[pic]

1. B

2. A

3. B

4. A

5. A

6. D

7. C

8. D

9. C

10. A

11. D

12. D

13. D

14. D

15. B

16. A

17. B

18. B

19. C

20. C

21. D

22. D

23. B

24. B

25. C

26. C

27. C

28. B

29. C

30. B

31. D

32. C

Explanations

1. The diameter of a circle is twice the radius. So, the diameter of the scale drawing is 6 cm. Set up a proportion, and solve for the diameter of the actual pond. Let x represent the diameter of the actual pond.

[pic]

Therefore, the diameter of the actual pond will be 12 feet.

2. The area of triangle A is 72 cm2. Since the dimensions of triangle A are two times larger than the dimensions of triangle B, the scale factor is 2.

To find the area of triangle B, first square the scale factor, 2.

22 = 4

Next, divide the area of triangle A by 4.

72 cm2 ÷ 4 = 18 cm2

3. First, set up a proportion to find the width of the image on the tablet.

[pic]

Next, convert 7.77 centimeters to millimeters.

[pic]

Therefore, the image should be 77.7 millimeters wide on the tablet.

4. Since the scale for the model on the left is 1 cm = 6 in, and the scale for the model on the right is 1 cm = 2 in, the area of the model on the right is 32, or 9, times larger than the area of the model on the left.

Divide the area of the model on the right by 9 to find the area of the model on the left.

108 sq cm ÷ 9 = 12 sq cm

5. Let x represent the length of the island on the blueprint. Set up a proportion and solve for x using cross multiplication.

[pic]

Let y represent the width of the island on the blueprint. Set up a proportion and solve for y using cross multiplication.

[pic]

Therefore, the dimensions of the island on the blueprint are [pic]inches by [pic]inch.

6. Since figure STUV and figure WXYZ are similar, their corresponding sides are proportional. So, side WZ corresponds to side SV.

Set up a proportion with the side lengths and the scale factor to solve for the length of side WZ.

[pic]

Therefore, the length of side WZ is 34 mm.

7. Since the scale for the model on the left is 1 in = 5 ft, and the scale for the model on the right is 1 in = 10 ft, the area of the model on the left is 22, or 4, times larger than the area of the model on the right.

Multiply the area of the model on the right by 4 to find the area of the model on the left.

9 sq in × 4 = 36 sq in

8. Since the scale for the model on the left is 1 cm = 5 ft, and the scale for the model on the right is 1 cm = 2.5 ft, the area of the model on the right is 22, or 4, times larger than the area of the model on the left.

Multiply the area of the model on the left by 4 to find the area of the model on the right.

21 sq cm × 4 = 84 sq cm

9. The scale for the microchip on the left is [pic].

The scale for the microchip on the right is [pic].

So, the microchip on the right is 2 times larger than the microchip on the left.

Multiply the height of the microchip on the left by 2 to find the height of the microchip on the right.

[pic]

10. The area of rectangle B is 18 cm2. Since the dimensions of rectangle A are four times larger than the dimensions of rectangle B, the scale factor is 4.

To find the area of rectangle A, first square the scale factor, 4.

42 = 16

Next, multiply the area of rectangle B by 16.

18 cm2 × 16 = 288 cm2

11. The area of triangle A is 90 cm2. Since the dimensions of triangle A are three times larger than the dimensions of triangle B, the scale factor is 3.

To find the area of triangle B, first square the scale factor, 3.

32 = 9

Next, divide the area of triangle A by 9.

90 cm2 ÷ 9 = 10 cm2

12. The area of rectangle A is 98 cm2. Since the dimensions of rectangle B are two times larger than the dimensions of rectangle A, the scale factor is 2.

To find the area of rectangle B, first square the scale factor, 2.

22 = 4

Next, multiply the area of rectangle A by 4.

98 cm2 × 4 = 392 cm2

13. Since the scale for the model on the left is 1 in = 12 ft, and the scale for the model on the right is 1 in = 3 ft, the model on the right is 4 times larger than the model on the left.

Multiply the side length of the model on the left by 4 to find the side length of the model on the right.

[pic]

14. First, set up a proportion to find the height of the demonstration calculator in centimeters.

[pic]

Next, convert 49 centimeters to meters.

[pic]

Therefore, the demonstration calculator is 0.49 meter tall.

15. Since the scale for the model on the left is 1 cm = 7.5 ft, and the scale for the model on the right is 1 cm = 3.75 ft, the model on the right is 2 times larger than the model on the left.

Divide the length of side b by 2 to find the length of side a.

10 cm ÷ 2 = 5 cm

16. Since the scale for the model on the left is 1 cm = 3 ft, and the scale for the model on the right is 1 cm = 9 ft, the area of the model on the left is 32, or 9, times larger than the area of the model on the right.

Divide the area of the model on the left by 9 to find the area of the model on the right.

72 sq cm ÷ 9 = 8 sq cm

17. Let x represent the actual length of the nightstand. Set up a proportion and solve for x using cross multiplication.

[pic]

Let y represent the actual width of the nightstand. Set up a proportion and solve for y using cross multiplication.

[pic]

Therefore, the actual dimensions of Trisha's nightstand are [pic]feet by [pic]feet.

18. The area of triangle A is 12.5 cm2. Since the dimensions of triangle B are three times larger than the dimensions of triangle A, the scale factor is 3.

To find the area of triangle B, first square the scale factor, 3.

32 = 9

Next, multiply the area of triangle A by 9.

12.5 cm2 × 9 = 112.5 cm2

19. The area of circle B is 324[pic] cm2. Since the dimensions of circle B are two times larger than the dimensions of circle A, the scale factor is 2.

To find the area of circle A, first square the scale factor, 2.

22 = 4

Next, divide the area of circle B by 4.

324[pic] cm2 ÷ 4 = 81[pic] cm2

20. Let x represent the length of the additional room on the blueprint. Set up a proportion and solve for x using cross multiplication.

[pic]

Let y represent the width of the additional room on the blueprint. Set up a proportion and solve for y using cross multiplication.

[pic]

Therefore, the dimensions of the additional room on the blueprint are 3 cm by 4 cm.

21. The scale for the television on the left is 1 cm = 6 in.

The scale for the television on the right is 1 cm = 2 in.

So, the television on the right is 3 times larger than the television on the left.

Multiply the diagonal of the screen in the television on the left by 3 to find the diagonal of the screen in the television on the right.

14 in × 3 = 42 in

22. Since the scale for the model on the left is 1 cm = 6.25 ft, and the scale for the model on the right is 1 cm = 1.25 ft, the area of the model on the right is 52, or 25, times larger than the area of the model on the left.

Multiply the area of the model on the left by 25 to find the area of the model on the right.

2 sq cm × 25 = 50 sq cm

23. First, convert 8 feet to inches.

[pic]

Next, set up a proportion to find the diagonal of the DVD cover art.

[pic]

Therefore, the diagonal of the DVD cover art is 8 inches.

24. The area of square A is 1,764 cm2. Since the dimensions of square A are three times larger than the dimensions of square B, the scale factor is 3.

To find the area of square B, first square the scale factor, 3.

32 = 9

Next, divide the area of square A by 9.

1,764 cm2 ÷ 9 = 196 cm2

25. Since the scale for the model on the left is 1 cm = 15 ft, and the scale for the model on the right is 1 cm = 7.5 ft, the model on the right is 2 times larger than the model on the left.

Multiply the height of the model on the left by 2 to find the height of the model on the right.

17 cm × 2 = 34 cm

26. The scale for the model on the left is 1 cm = 3.75 m, and the scale for the model on the right is 1 cm = 2.5 m. Divide to find how much larger the model on the right is than the model on the left.

3.75 ÷ 2.5 = 1.5

So, the model on the right is 1.5 times larger than the model on the left.

Multiply the length of side a by 1.5 to find the length of side b.

4 cm × 1.5 = 6 cm

27. Since figure KLMN and figure PQRS are similar, their corresponding sides are proportional. Sides KN and MN correspond to sides PS and RS, respectively.

|KN = |6 cm | |PS = |12 cm |

|MN = |14 cm | |RS = |28 cm |

Notice that the lengths of sides KN and MN can be multiplied by 2 to get the length of sides PS and RS, respectively.

Therefore, the scale factor of figure KLMN to figure PQRS is 2.

28. The area of triangle B is 60 cm2. Since the dimensions of triangle B are two times larger than the dimensions of triangle A, the scale factor is 2.

To find the area of triangle A, first square the scale factor, 2.

22 = 4

Next, divide the area of triangle B by 4.

60 cm2 ÷ 4 = 15 cm2

29. The scale for the model on the left is 1 in = 3 ft, and the scale for the model on the right is 1 in = 8.25 ft. Divide to find how much larger the model on the left is than the model on the right.

8.25 ÷ 3 = 2.75

So, the model on the left is 2.75 times larger than the model on the right.

Divide the width of the model on the left by 2.75 to find the width of the model on the right.

8.25 in ÷ 2.75 = 3 in

30. Write the ratios of the blueprint dimensions to the actual dimensions, and then reduce the fractions.

[pic]

Therefore, the scale of the blueprint is 1 centimeter = 3 feet.

31. The area of rectangle B is 72 cm2. Since the dimensions of rectangle A are three times larger than the dimensions of rectangle B, the scale factor is 3.

To find the area of rectangle A, first square the scale factor, 3.

32 = 9

Next, multiply the area of rectangle B by 9.

72 cm2 × 9 = 648 cm2

32. Since the scale for the diagram on the left is 1 in = 12 ft, and the scale for the diagram on the right is 1 in = 4 ft, the diagram on the right is 3 times larger than the diagram on the left.

Multiply the length of the diagram on the left by 3 to find the length of the diagram on the right.

6 in × 3 = 18 in

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