Similar figures and length ratios

(a) 15 cm

1. Scale factor $= 10 \div 4 = 2.5$.
2. $$6 \times 2.5 = 15$$

• M1 Finding the scale factor 2.5 (or the ratio $6 : 4$ as 1.5).
• A1 Correct answer: 15 cm.

(a) 12 m

1. The sun's rays make similar right-angled triangles.
2. Height is $\frac{1.8}{2.4} = 0.75$ times the shadow.
   $$16 \times 0.75 = 12$$

• P1 A correct ratio: $\frac{16}{2.4}$ (scale factor) or $\frac{1.8}{2.4}$ (height per metre of shadow).
• P1 Applying it: $1.8 \times \frac{16}{2.4}$ or $16 \times 0.75$.
• A1 Correct answer: 12 m.

(a) $12$ cm

1. $$8/5$$
2. $$7.5\times 8/5$$
3. The length scale factor from ABC to DEF is DE/AB = 8/5.
4. DF corresponds to AC, so DF = 7.5 $\times$ 8/5 = 12 cm.

• P1 Establishing $8/5$ or an equivalent valid method.
• P1 Establishing $7.5\times 8/5$ or an equivalent valid method.
• A1 Correct answer: $12$ cm

(a) $14$ cm

1. Find the scale factor from ABC to DEF using corresponding sides.
   $$7/4$$
2. Apply the same factor to BC.
   $$8\times (7/4)$$
3. Therefore $14$ cm.

• M1 Find the scale factor from ABC to DEF using corresponding sides.
• M1 Apply the same factor to BC.
• A1 Correct answer: $14$ cm

(b) $12$ cm

1. Use the inverse factor to return to the smaller triangle.
   $$21\times (4/7)$$
2. Therefore $12$ cm.

• M1 Use the inverse factor to return to the smaller triangle.
• A1 Correct answer: $12$ cm

(a) $20$ cm

1. The ratio of corresponding widths also applies to the heights.
   $$32\times 5/8$$
2. Therefore $20$ cm.

• M1 The ratio of corresponding widths also applies to the heights.
• A1 Correct answer: $20$ cm

(b) $50$ cm

1. Add two widths and two heights.
   $$2\times (5+20)$$
2. Therefore $50$ cm.

• M1 Add two widths and two heights.
• A1 Correct answer: $50$ cm

(a) $18$ cm

1. Perimeters scale by the same factor as all corresponding lengths.
   $$12\times 54/36$$
2. Therefore $18$ cm.

• M1 Perimeters scale by the same factor as all corresponding lengths.
• A1 Correct answer: $18$ cm

(b) No. The question gives similarity. Perimeters alone do not establish equal angles or proportional corresponding sides.

1. No. The question gives similarity. Perimeters alone do not establish equal angles or proportional corresponding sides.

• C1 Correct conclusion with supporting reasoning: No. The question gives similarity. Perimeters alone do not establish equal angles or proportional corresponding sides.

(a) $2.5$

1. Divide the larger height by the smaller height.
   $$15/6$$
2. Therefore $2.5$.

• M1 Divide the larger height by the smaller height.
• A1 Correct answer: $2.5$

(b) $6$ cm

1. Multiply the smaller radius by the scale factor.
   $$2.4\times 2.5$$
2. Therefore $6$ cm.

• M1 Multiply the smaller radius by the scale factor.
• A1 Correct answer: $6$ cm

(a) $3$

1. Compare the perimeters, which scale by the same factor as the sides.
   $$60/20$$
2. Therefore $3$.

• M1 Compare the perimeters, which scale by the same factor as the sides.
• A1 Correct answer: $3$

(b) $21$ cm

1. Multiply the longer side of A by the scale factor.
   $$7\times 3$$
2. Therefore $21$ cm.

• M1 Multiply the longer side of A by the scale factor.
• A1 Correct answer: $21$ cm