Scale Drawings and Similar Figures: practice solutions, Grade 7 – download the PDF

1 Drawing to real length ★★★

Each centimeter stands for 4 m, so the actual length is \( 6 \times 4 = 24 \) m.

The wall is 24 m long.

2 Find the scale factor ★★★

\( \dfrac{12}{4} = 3 \) and \( \dfrac{18}{6} = 3 \). The two ratios agree.

The scale factor is 3.

3 A road map ★★★

\( 3.5 \times 10 = 35 \).

The cities are 35 mi apart.

4 True or false: angles ★★★

No. In a scale drawing the lengths change but the angles stay the same. A \( 40^\circ \) angle is still \( 40^\circ \) after the enlargement.

Mia is wrong.

5 Real to drawing ★★★

\( 50 \div 5 = 10 \).

The field is 10 cm wide in the drawing.

6 A model car ★★★

The real car is \( 7 \times 24 = 168 \) in long. Then \( 168 \div 12 = 14 \) ft.

The real car is 14 ft long.

7 Complete the table ★★★

Multiply by 3 to go from the drawing to the actual length: \( 2 \times 3 = 6 \) and \( 5 \times 3 = 15 \). Divide by 3 for the last row: \( 45 \div 3 = 15 \).

8 A bedroom floor plan ★★★

1. \( 2.5 \times 4 = 10 \) ft and \( 3 \times 4 = 12 \) ft.
2. \( 10 \times 12 = 120 \). The bedroom is 10 ft by 12 ft, with an area of 120 square feet.

9 Similar triangles ★★★

Scale factor: \( \dfrac{9}{6} = 1.5 \). Check with the second pair: \( \dfrac{12}{8} = 1.5 \).

\( x = 10 \times 1.5 = 15 \). The third side is 15.

10 Kilometers on a map ★★★

The actual length is \( 7.2 \times 500 = 3600 \) m. Since 1000 m = 1 km, \( 3600 \div 1000 = 3.6 \) km.

The trail is 3.6 km long.

11 Fractional scale ★★★

Each \( \dfrac{1}{4} \) in = 0.25 in stands for 1 ft. The number of quarter inches in 3.5 in is \( 3.5 \div 0.25 = 14 \).

The porch is 14 ft long.

12 Changing the scale ★★★

The actual size is \( 8 \times 2 = 16 \) m by \( 5 \times 2 = 10 \) m.

New drawing: \( 16 \div 4 = 4 \) cm and \( 10 \div 4 = 2.5 \) cm.

The new drawing is 4 cm by 2.5 cm. (Shortcut: the factor is \( \dfrac{2}{4} = \dfrac{1}{2} \).)

13 Is it a scale drawing? ★★★

\( \dfrac{6}{4} = 1.5 \) but \( \dfrac{8}{6} \approx 1.33 \). The ratios are not equal.

No, it is not a scale drawing, because the sides are not proportional.

14 A model of a tower ★★★

150 m = 15,000 cm. Then \( 15,000 \div 200 = 75 \).

The model is 75 cm tall.

15 Area of an enlargement ★★★

1. Original: \( 3 \times 5 = 15 \) cm². The enlargement is 12 cm by 20 cm, so its area is \( 12 \times 20 = 240 \) cm².
2. \( 240 \div 15 = 16 = 4^2 \). The area was multiplied by 16.

16 Area of a lake ★★★

Map area: \( 2.4 \times 1.5 = 3.6 \) cm². Each cm² stands for \( 5^2 = 25 \) km².

\( 3.6 \times 25 = 90 \). The lake covers 90 km².

Check: the actual sides are 12 km and 7.5 km and \( 12 \times 7.5 = 90 \).

17 A garden plan ★★★

Each cm² stands for \( 50^2 = 2500 \) cm². The actual area is \( 48 \times 2500 = 120,000 \) cm².

Since 1 m² = 10,000 cm², \( 120,000 \div 10{,}000 = 12 \). The garden has an area of 12 m².

18 Find the scale factor from areas ★★★

The area ratio is \( \dfrac{81}{36} = \dfrac{9}{4} \). Since area ratio \( = k^2 \), \( k = \sqrt{\dfrac{9}{4}} = \dfrac{3}{2} = 1.5 \).

Copy: \( 9 \times 1.5 = 13.5 \) in by \( 4 \times 1.5 = 6 \) in. Check: \( 13.5 \times 6 = 81 \).

The scale factor is 1.5 and the copy is 13.5 in by 6 in.

19 Photo enlargement ★★★

1. \( \dfrac{10}{4} = 2.5 \).
2. \( 6 \times 2.5 = 15 \) in.
3. The area goes from \( 4 \times 6 = 24 \) in² to \( 10 \times 15 = 150 \) in², and \( 150 \div 24 = 6.25 = 2.5^2 \). The area is 6.25 times larger.

20 Floor of a model house ★★★

1. \( 24 \times 50 = 1200 \) cm = 12 m and \( 18 \times 50 = 900 \) cm = 9 m.
2. \( 12 \times 9 = 108 \). The actual floor area is 108 m².

Check: the model area is \( 24 \times 18 = 432 \) cm², and \( 432 \times 2500 = 1,080,000 \) cm² = 108 m².

21 Perimeter and area of similar triangles ★★★

1. \( 6 \times 2.5 = 15 \), \( 8 \times 2.5 = 20 \), \( 10 \times 2.5 = 25 \).
2. ABC: \( 6 + 8 + 10 = 24 \). DEF: \( 15 + 20 + 25 = 60 \). \( 60 \div 24 = 2.5 \): the perimeter is multiplied by 2.5.
3. ABC: \( \dfrac{6 \times 8}{2} = 24 \). DEF: \( \dfrac{15 \times 20}{2} = 150 \). \( 150 \div 24 = 6.25 = 2.5^2 \): the area is multiplied by 6.25.

22 Which scale is bigger? ★★★

Map X: \( 10 \div 2 = 5 \) cm. Map Y: \( 10 \div 5 = 2 \) cm.

The scale factor from Y to X is \( \dfrac{5}{2} = 2.5 \). Map X shows the park 2.5 times longer (and its area 6.25 times larger).