Enlargement and similarity in 2D

Unit 11 Enlargement and similarity in 2D

Key unit competence: Apply the principles of similarity and enlargement in solving problems involving scale, proportion and transformation in two dimensions.

Key ideas

• Similar figures have the same shape: corresponding angles are equal and corresponding sides are in the same ratio $k$ (the scale factor).
• Triangles are similar if two angles are equal (AA), or the three pairs of sides are proportional (SSS), or two sides are proportional with the included angle equal (SAS).
• An enlargement with centre $O$ and scale factor $k$ maps $P$ to $P^{\prime }$ with $\overrightarrow{O{P}^{\prime }}=k\overrightarrow{OP}$. Centre at the origin: $(x,y)\mapsto (kx,ky)$; centre $C(a,b)$: $P^{\prime }=C+k(P-C)$.
• If $|k|>1$ the image is bigger; if $0<|k|<1$ smaller; if $k<0$ the image is on the other side of the centre (inverted).
• Lengths are multiplied by $|k|$, areas by $k^2$ and volumes by $|k{|}^3$. Angles are unchanged.
• To find the centre: join each point to its image; the lines meet at the centre. $k=\frac{\text{image length}}{\text{object length}}$.

Formulas to know

Common misconceptions

• "If lengths double, area doubles." No: area is multiplied by $2^2=4$.
• "An enlargement always makes figures bigger." No: if $0<|k|<1$ it makes them smaller.

Worked examples

Practice questions

Easier and harder questions are mixed, just like in a real exam.

1. Enlarge the square $O(0,0)$, $A(2,0)$, $B(2,2)$, $C(0,2)$ with centre $O$ and $k=-1.5$. Give the image and its area.

   Show answer

   $O(0,0)$, $A^{\prime }(-3,0)$, $B^{\prime }(-3,-3)$, $C^{\prime }(0,-3)$; area $9$ (original 4 times $2.25$).

   Common mistake: Ignoring the negative sign.

2. Explain why all circles are similar.

   Show answer

   Any circle can be mapped to any other by a translation (centre to centre) and an enlargement with $k=\frac{r_2}{r_1}$.

   Common mistake: Saying circles are similar only if they have the same centre.

3. Show that triangles with sides 5, 7, 9 and 10, 14, 18 are similar and give the scale factor.

   Show answer

   $\frac{10}{5}=\frac{14}{7}=\frac{18}{9}=2$. SSS similarity, $k=2$.

   Common mistake: Comparing non-corresponding sides.

4. A student 1.6 m tall stands 4 m from a lamp post and casts a 2 m shadow. How tall is the lamp post?

   Show answer

   Similar triangles: $\frac{h}{6}=\frac{1.6}{2}$, $h=4.8$ m.

   Common mistake: Using 4 m instead of $4+2=6$ m.

5. Two similar jugs have heights 15 cm and 20 cm. The smaller holds 0.54 L. How much does the larger hold?

   Show answer

   $k=\frac{4}{3}$, $k^3=\frac{64}{27}$; $0.54\times \frac{64}{27}=1.28$ L.

   Common mistake: Multiplying by $k$ or $k^2$.

6. The ratio of areas of two similar triangles is $25:49$. A side of the smaller is 10 cm. Find the corresponding side of the larger.

   Show answer

   $k=\frac{7}{5}$; side $=14$ cm.

   Common mistake: Using $\frac{49}{25}\times 10$.

7. Find the image of $P(5,1)$ under the enlargement with centre $C(1,3)$ and scale factor $\frac{1}{2}$.

   Show answer

   $C+\frac{1}{2}(P-C)=(1,3)+\frac{1}{2}(4,-2)=(3,2)$.

   Common mistake: Applying the scale factor to $P$ directly.

8. $A(1,2)$ maps to $A^{\prime }(3,6)$ and $B(2,1)$ maps to $B^{\prime }(6,3)$. Find the centre and scale factor.

   Show answer

   Both images are 3 times the objects: centre $O(0,0)$, $k=3$.

   Common mistake: Using $k=2$ from $3-1$.

Matching and drag-and-drop

1. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. Lengths are multiplied by 3. Areas are multiplied by:	a. 540 cm³
   2. Similar triangles have sides 3, 4, 5 and $6$, $8$, $x$. Find $x$.	b. 9
   3. Two similar cylinders have heights 6 cm and 9 cm. The smaller has volume 160 cm³. Find the volume of the larger.	c. $(-2,1)$
   4. Enlarge $(4,-2)$ with centre $O$ and scale factor $-\frac{1}{2}$.	d. 10

   Show answer

   1 → b (9) · 2 → d (10) · 3 → a (540 cm³) · 4 → c ($(-2,1)$)

   • $k^2$.
   • $k=2$.
   • Step 1. Length ratio $\frac{9}{6}=\frac{3}{2}$, so volume ratio ${\left(\frac{3}{2}\right)}^3=\frac{27}{8}$.
     Step 2. $160\times \frac{27}{8}=540$ cm³.
   • Multiply by $-\frac{1}{2}$.

2. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   $1:9$  ·  0.5 km²  ·  8  ·  $1:27$
   1. A map has scale 1 : 25 000. A forest covers 8 cm² on the map. Its real area in km² is .
   2. Lengths are multiplied by 2. Volumes are multiplied by .
   3. The surface areas of two similar spheres are in the ratio $1:9$. Their volumes are in ratio .
   Show answer

   1: 0.5 km² · 2: 8 · 3: $1:27$

   • Step 1. 1 cm ↔ 250 m, so 1 cm² ↔ 62 500 m².
     Step 2. $8\times 62500=500000$ m².
   • $k^3$.
   • Step 1. $k=3$.
     Step 2. So volumes $1:27$.

3. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. Triangle ABC is similar to triangle PQR with $AB=6$, $PQ=9$, $BC=8$. Find $QR$.	a. 12
   2. Two similar triangles have areas in the ratio $4:9$. A side of the smaller is 6 cm. Find the matching side of the larger.	b. $(4,6)$
   3. On a map of scale $1:25000$, a field has area 8 cm². Find its real area.	c. 9 cm
   4. Enlarge $(2,3)$ with centre $O$ and scale factor 2.	d. 0.5 km²

   Show answer

   1 → a (12) · 2 → c (9 cm) · 3 → d (0.5 km²) · 4 → b ($(4,6)$)

   • $k=1.5$.
   • Step 1. Length ratio $=\sqrt{\frac{9}{4}}=\frac{3}{2}$.
     Step 2. $6\times \frac{3}{2}=9$ cm.
   • Step 1. Area scale: $25000^2=6.25\times 10^8$; real area $=8\times 6.25\times 10^8=5\times 10^9$ cm².
     Step 2. $1$ km² $=10^{10}$ cm², so the area is $0.5$ km².
   • Multiply both coordinates by 2.

4. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   45 cm  ·  20 cm  ·  4 cm²  ·  35 cm
   1. A map has scale factor $\frac{1}{50000}$. A real area of 1 km² appears on the map as .
   2. A photo 10 cm by 15 cm is enlarged to width 30 cm. Its new length is .
   3. A model car is made at scale 1 : 20. The real car is 4 m long. The model is .
   Show answer

   1: 4 cm² · 2: 45 cm · 3: 20 cm

   • 1 km $=100000$ cm $\to$ 2 cm on the map, so $2\times 2$.
   • $k=3$.
   • $400÷20$.

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