Similar Shapes

Divide a length on B by the matching length on A.

Work out the scale factor from AB to DE first.

Compare the heights to find how many times smaller the small triangle is.

Find the number you multiply 4 by to get 6.

In similar shapes every length is multiplied by the same number. Adding the same number does not work.

Multiply the length on P by the scale factor. A scale factor less than 1 makes the shape smaller.

1. Lengths on Q are the matching lengths on P multiplied by \frac{3}{4}.
2. 20\div4=5, and 5\times3=15.
3. The matching side on Q is 15 cm.

Use the arcs to match the angles. Then match each side by the two angles at its ends.

1. PQ matches AB, so the scale factor is 10\div4=2.5.
2. QR matches BC, which is 6 cm.
3. x=6\times2.5=15.

Angle DFE matches angle ACB. Do the angles change when a shape is made bigger?

1. Angle ACB =180-75-48=57°.
2. Angle DFE matches angle ACB.
3. In similar shapes the matching angles are equal: the lengths double but the angles stay the same.
4. Angle DFE = 57°.

Divide each length of Q by the matching length of P. Are the answers the same?

1. Short sides: 9\div6=1.5.
2. Long sides: 13\div8=1.625.
3. The two scale factors are different, so the rectangles are not similar.

Find the scale factor from AB to DE first, then use it for both missing sides.

Part (a)

1. The scale factor is 9\div6=1.5.
2. EF matches BC, so x=8\times1.5=12.

Part (b)

1. DF matches AC, so y=5\times1.5=7.5.

The two bases, BC and QR, match. Use them to find the scale factor.

1. BC matches QR, so the scale factor from ABC to PQR is 10\div6=\frac{5}{3}.
2. PQ matches AB, which is 9 cm.
3. 9\div3=3 and 3\times5=15, so PQ = 15 cm.

Change 4.5 m into centimetres before you compare it with the model.

1. 4.5 m = 450 cm.
2. Scale factor from model to car =450\div15=30.
3. Height of the real car =5\times30=150 cm.
4. 150 cm = 1.5 m.

Find the width of rectangle B first.

1. Scale factor =15\div6=2.5.
2. Width of B =4\times2.5=10 cm.
3. Perimeter of B =10+15+10+15=50 cm.

Shape Q is the bigger shape. The 5 parts of the ratio stand for 30 cm.

1. The 5 parts for Q stand for 30 cm, so 1 part is 30\div5=6 cm.
2. Shape P has 2 parts: 2\times6=12 cm.
3. The matching side on P is 12 cm.

In similar shapes, are the lengths multiplied by a number or added to?

Part (a)

1. In similar shapes, every length is multiplied by the same scale factor.
2. Ravi added 2 cm, which does not keep the shape the same.

Part (b)

1. Scale factor =6\div4=1.5.
2. EF =6\times1.5=9 cm.

Work out the perimeter of A. Perimeters are lengths, so they change by the same scale factor as the sides.

1. Perimeter of A =6+9+10+15=40 cm.
2. Scale factor from A to B =60\div40=1.5.
3. The longest side of A is 15 cm, so the longest side of B is 15\times1.5=22.5 cm.

Compare the two shadows to find the scale factor. The heights match each other.

1. The shadows match: scale factor =6.4\div0.8=8.
2. The heights match, so h=1.2\times8=9.6.
3. The tree is 9.6 m tall.

Find the height of photo B first.

1. Scale factor from A to B =12\div10=1.2.
2. Height of B =15\times1.2=18 cm.
3. Height of C =18\times2.5=45 cm.

If the side on A is 1 part, the side on B is 3 parts. How many parts is the difference?

Part (a)

1. The side on A is 1 part and the side on B is 3 parts.
2. The difference is 3-1=2 parts, and 2 parts are 14 cm.
3. 1 part =14\div2=7 cm, so the side on A is 7 cm.

Part (b)

1. 7\times3=21 cm. Check: 21-7=14 cm.

For each size, divide the width by 60 and the length by 90. Similar flags give the same answer both times.

Part (a)

1. 90 by 120: 30 cm has been added to each length. 90\div60=1.5 but 120\div90 is about 1.33, so it is not similar.
2. 120 by 160: 120\div60=2 but 160\div90 is about 1.78, so it is not similar.
3. 80 by 130: 80\div60 is about 1.33 but 130\div90 is about 1.44, so it is not similar.
4. 100 by 150: 100\div60 and 150\div90 both give about 1.67, the same scale factor.
5. So the 100 cm by 150 cm flag is similar.

Part (b)

1. In the original flag, the length is 90\div60=1.5 times the width.
2. So the new length =140\times1.5=210 cm.