Practice Maths

Similar Figures and Scale Factors

Key Ideas

Key Terms

similar
(~) if they have the same shape but (possibly) different sizes.
corresponding angles are equal
And corresponding sides are in the same ratio.
scale factor
K = (image side length) ÷ (original side length). If k > 1 the image is larger; if 0 < k < 1 it is smaller.
Similarity tests for triangles

   – AA: two pairs of equal angles (the third follows automatically).
   – SAS similarity: two pairs of sides in the same ratio AND the included angles equal.
   – SSS similarity: all three pairs of corresponding sides in the same ratio.
Hot Tip Congruent figures are a special case of similar figures where k = 1. Similar does NOT mean congruent unless all side lengths are also equal.

Worked Example

Question: ▵ABC has sides 6, 8, and 10. ▵PQR has sides 9, 12, and 15. Are these triangles similar? If so, find the scale factor from ▵ABC to ▵PQR.

Step 1 — Check if all side ratios are equal.
9 ÷ 6 = 1.5     12 ÷ 8 = 1.5     15 ÷ 10 = 1.5

Step 2 — Conclude.
All ratios equal 1.5, so ▵ABC ~ ▵PQR by SSS similarity.

Step 3 — State the scale factor.
Scale factor from ▵ABC to ▵PQR = 1.5 (or 3/2).
This means every side of ▵PQR is 1.5 times the corresponding side of ▵ABC.

What Makes Two Figures Similar?

Two figures are similar if they have exactly the same shape but can be different sizes. One is an enlargement (or reduction) of the other. The formal definition requires two conditions to hold simultaneously:

  1. All corresponding angles are equal.
  2. All corresponding sides are in the same ratio (proportional).

Real-world examples of similar figures: a photo and an enlargement of it, a map and the real landscape it represents, scale models of buildings, architectural drawings.

Identifying Corresponding Parts

When two figures are stated to be similar, their vertices are listed in matching order. For example, if ▵ABC ~ ▵DEF, then:

  • A corresponds to D, B corresponds to E, C corresponds to F
  • ∠A = ∠D, ∠B = ∠E, ∠C = ∠F
  • AB/DE = BC/EF = AC/DF (all equal to the scale factor)

Always match vertices by order, not by appearance. In exam questions, the similarity statement tells you exactly which parts correspond.

The Scale Factor

The scale factor k is the ratio by which all lengths change from one figure to the other:

k = (length in new figure) ÷ (corresponding length in original figure)

  • If k > 1, the new figure is an enlargement.
  • If 0 < k < 1, the new figure is a reduction.
  • If k = 1, the figures are congruent (identical size and shape).

The scale factor applies to all corresponding lengths, but areas scale by k2 (an important extension concept).

Testing for Similarity

To check if two triangles are similar without being told, you need only one of these conditions:

  • AA (Angle-Angle): Two pairs of corresponding angles are equal (the third pair must also match since triangle angles sum to 180°).
  • SSS similarity: All three pairs of sides are in the same ratio.
  • SAS similarity: Two pairs of sides in the same ratio and the included angles are equal.
Key tip: When writing a similarity statement, the order of vertices matters — always list them in corresponding order. ▵ABC ~ ▵DEF means A corresponds to D. Getting the order wrong will lead to incorrect side ratios and wrong answers in all subsequent calculations.

Mastery Practice

  1. State whether each pair of shapes must be similar, giving a reason. Fluency

    1. Two squares with different side lengths.
    2. Two rectangles: one is 4 cm × 6 cm, the other is 6 cm × 9 cm.
    3. Two equilateral triangles with different side lengths.
    4. Two right-angled triangles, both with a 35° angle.
    5. Two isosceles triangles, each with an apex angle of 50°.
    6. Two regular hexagons.

  2. Calculate the scale factor from the first figure to the second. State whether it is an enlargement or reduction. Fluency

    1. Sides 4 cm and 10 cm.
    2. Sides 15 cm and 5 cm.
    3. Sides 7 cm and 10.5 cm.
    4. Sides 20 cm and 8 cm.
    5. Rectangle 3 cm × 5 cm enlarged to 7.5 cm × 12.5 cm.
    6. Triangle with sides 12, 16, 20 and corresponding triangle with sides 9, 12, 15.
    7. Pentagon with sides all 6 cm and a similar pentagon with sides all 4 cm.
    8. Square with side 9 cm and similar square with side 12 cm.
    9. Triangle with sides 5, 12, 13 and similar triangle with sides 10, 24, 26.
    10. Figure with perimeter 48 cm and similar figure with perimeter 36 cm.

  3. For each pair of triangles, determine whether they are similar. If similar, name the test (AA, SAS, SSS) and write the similarity statement with vertices in matching order. Understanding

    1. ▵ABC with ∠A = 50°, ∠B = 70° and ▵PQR with ∠P = 50°, ∠R = 60°.
    2. ▵DEF with DE = 4, EF = 6, ∠E = 80° and ▵GHI with GH = 6, HI = 9, ∠H = 80°.
    3. ▵JKL with sides 3, 5, 7 and ▵MNO with sides 6, 10, 14.
    4. ▵PQR with PQ = 8, QR = 12 and ▵STU with ST = 6, TU = 9 and ∠Q = ∠T = 55°.
    5. ▵ABC with ∠B = 90°, AB = 6, BC = 8 and ▵XYZ with ∠Y = 90°, XY = 9, YZ = 12.

  4. Write a formal proof to show that each pair of triangles is similar. Understanding

    1. In the diagram, AB ∥ DE. C is the intersection of AE and BD. Prove ▵ABC ~ ▵EDC.
    2. In ▵PQR, a line ST is drawn parallel to QR, cutting PQ at S and PR at T. Prove ▵PST ~ ▵PQR.
    3. Two triangles share a common angle at A. AB = 4, AC = 6, AD = 8, AE = 12 (with D on AB extended and E on AC extended). Prove ▵ABC ~ ▵ADE.

  5. Real-world similarity and scale. Problem Solving

    1. A photograph is 10 cm wide and 15 cm tall. It is enlarged so that the width becomes 25 cm. Find the new height, giving a reason.
    2. A tree casts a shadow of 8 m at the same time that a 1.5 m tall person casts a shadow of 2 m. Explain why the triangles formed are similar, then find the height of the tree.
    3. A model car is built to a scale of 1 : 24. The real car is 4.32 m long. Find the length of the model car in centimetres.
    4. In a right-angled triangle, the altitude from the right angle to the hypotenuse creates two smaller triangles. Show that each smaller triangle is similar to the original large triangle, naming the similarity test used.
  6. Find the unknown side. Two similar figures are shown. Use the scale factor to find the value of x in each case.
    Problem Solving
    1. Two similar rectangles: the first has length 10 cm and width 6 cm. The second has length 15 cm and width x cm. Find x.
    2. Two similar triangles: the first has sides 8 cm, 12 cm, and 16 cm. The second has its shortest side equal to 6 cm. Find the other two sides.
    3. A photograph is 12 cm × 8 cm. It is reduced so the longer side becomes 9 cm. Find the new shorter side.
    4. Two similar pentagons: corresponding sides are 5 cm and 8 cm. The original pentagon has a perimeter of 35 cm. Find the perimeter of the larger pentagon.
  7. Area and perimeter ratios. Use the properties of similar figures to solve these problems.
    Problem Solving
    1. Two similar triangles have a scale factor of 3. The perimeter of the smaller triangle is 24 cm. Find the perimeter of the larger triangle.
    2. Two similar squares have side lengths 4 cm and 10 cm. Find the ratio of their areas. Then find the area of the larger square.
    3. A garden is photographed and the scale factor from the photo to the real garden is 1 : 50. In the photo, a rectangular section is 6 cm × 4 cm. Find the actual area of the section in m².
  8. Maps and models. Apply similarity and scale factors in real-world contexts.
    Problem Solving
    1. A map has a scale of 1 : 250 000. Two cities are 8 cm apart on the map. Find the actual distance in kilometres.
    2. An architect draws a floor plan at a scale of 1 : 100. A room on the plan measures 5.2 cm × 3.8 cm. Find the actual dimensions of the room in metres and the actual floor area in m².
    3. A model aeroplane is built at a scale of 1 : 72. The model has a wingspan of 18.5 cm. Calculate the actual wingspan of the aeroplane in metres, rounding to 2 decimal places.
    4. On a map scaled at 1 : 20 000, a lake has an area of 12 cm². What is the actual area of the lake in m²?
  9. Indirect measurement. Use similar triangles and scale factors to find lengths that cannot be measured directly.
    Problem Solving
    1. A student wants to find the width of a river. She marks out two similar triangles on the bank. The sides of the triangles are in the ratio 1 : 4, and the measurable side is 6.5 m. Find the width of the river.
    2. A vertical pole casts a shadow of 3 m. At the same time, a nearby building casts a shadow of 18 m. The pole is 2 m tall. Find the height of the building.
    3. A 1.8 m tall person stands 5 m from a lamp post. The tip of their shadow (cast by the lamp post light) is 2 m behind them. Set up a pair of similar triangles and find the height of the lamp post.
  10. Multi-step reasoning. These questions require you to first prove similarity, then use it to find unknown quantities.
    Problem Solving
    1. In a triangle PQR, a line XY is drawn parallel to QR, with X on PQ and Y on PR. PX = 4 cm, XQ = 6 cm, PY = 5 cm. (a) Prove that ▵PXY ~ ▵PQR. (b) Find YR. (c) Find the ratio of the perimeters of ▵PXY and ▵PQR.
    2. Two similar cylinders have radii in the ratio 2 : 5. (a) Write the scale factor from the smaller to the larger cylinder. (b) If the smaller cylinder has a volume of 32π cm³ and the volume formula scales as k³ for similar 3D shapes, find the volume of the larger cylinder in terms of π.
    3. A company produces two similar tins. The smaller tin has a height of 8 cm and holds 400 mL. The larger tin has a height of 12 cm. Using the cube of the scale factor for volume, find the capacity of the larger tin to the nearest mL.
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