Practice Maths

Finding Unknown Sides in Similar Figures

Key Ideas

Key Terms

corresponding sides are in the same ratio
(the scale factor k).
Perimeter scale factor = k
If sides scale by k, so does the perimeter.
Area scale factor = k²
If linear dimensions scale by k, the area scales by k².
Hot Tip Label corresponding sides before writing any ratio. If ▵ABC ~ ▵DEF, then AB corresponds to DE, BC to EF, and CA to FD. The ratio AB/DE = BC/EF = CA/FD = k.

Worked Example

Question: Two similar triangles have sides 6, 8, 10 and 9, x, y. Find x and y.

Step 1 — Find the scale factor:   k = 9 ÷ 6 = 1.5

Step 2 — Find x:   x = 8 × 1.5 = 12

Step 3 — Find y:   y = 10 × 1.5 = 15

Check:   9 : 12 : 15 = 3 : 4 : 5, which is the same ratio as 6 : 8 : 10 = 3 : 4 : 5. ✓

Setting Up the Proportion

When two figures are similar, all corresponding sides are in the same ratio (the scale factor). To find an unknown side, set up a proportion using a matching pair of known sides:

(unknown side) / (corresponding known side) = (one known side) / (corresponding known side in the other figure)

It helps to write both ratios in the same order: always "new figure side / original figure side" or always "original / new". Mixing the order is the most common source of errors.

Method 1: Using the Scale Factor

Step 1: Find the scale factor k by dividing one known corresponding pair: k = (side in figure 2) / (side in figure 1).

Step 2: Multiply every known side in figure 1 by k to get the corresponding side in figure 2 (or divide to go the other direction).

This method is fastest when the scale factor is a clean number (like 2 or 0.5).

Method 2: Cross-Multiplying the Proportion

Write the proportion as a fraction equation and cross-multiply to solve:

x / 9 = 8 / 12 ⇒ 12x = 72 ⇒ x = 6

Cross-multiplication is reliable even when the scale factor is not a whole number. Always check that your answer is reasonable: if figure 2 is larger than figure 1, its sides should all be larger too.

Common Mistakes to Avoid

  • Mismatching sides: Always identify which sides correspond before setting up the ratio. The longest side corresponds to the longest side, etc. If the figures are in different orientations, rotate mentally or use the similarity statement to match correctly.
  • Mixing up the ratio order: Keep "new/old" or "old/new" consistent on both sides of the proportion.
  • Forgetting to check: After solving, substitute back and verify the ratios are equal.

Multi-Step Similarity Problems

Some problems involve a chain of similar triangles: triangle A is similar to triangle B, and triangle B is similar to triangle C. Solve each similarity pair in turn, using the answer from one step as input to the next. Label each step clearly and don't round until the very end.

Exam tip: Always state the similarity (e.g., "▵ABC ~ ▵DEF, AA") and then write the proportion before solving. Examiners need to see that you have correctly matched the corresponding sides — a bare calculation with no setup can lose method marks even if the answer is correct.

Mastery Practice

  1. Find the value of x in each pair of similar figures. Fluency

    1. Similar triangles: sides 4 cm and 6 cm are corresponding. The side corresponding to 10 cm is x. Find x.
    2. Similar rectangles: one has length 8 m, the similar one has corresponding length 12 m. The width of the first is 5 m. Find the width x of the second.
    3. ▵PQR ~ ▵STU with PQ = 7 cm, ST = 10.5 cm. If QR = 9 cm, find TU.
    4. Two similar pentagons: one has a side of 6 cm and the similar one has the corresponding side of 4 cm. Another side of the first is 9 cm. Find the corresponding side x.
    5. Similar triangles: sides 15 and 10 are corresponding. A side of the smaller triangle is 8. Find the corresponding side of the larger triangle.

  2. ▵ABC ~ ▵DEF. AB = 6, BC = 9, CA = 12, DE = 4. Find EF and FD. Fluency

    1. Find the scale factor k from ▵ABC to ▵DEF.
    2. Find EF.
    3. Find FD.
    4. Find the perimeter of each triangle.
    5. What is the ratio of the perimeters? How does this relate to k?

  3. Using the area scale factor. Fluency

    1. Two similar triangles have a scale factor of k = 3. If the smaller triangle has area 8 cm², find the area of the larger triangle.
    2. Two similar rectangles have areas 50 cm² and 200 cm². Find the scale factor k (from smaller to larger).
    3. Two similar figures have a scale factor of 2.5. If the larger figure has area 125 m², find the area of the smaller figure.
    4. A photo is enlarged by a scale factor of 4. If the original area is 24 cm², what is the area of the enlargement?

  4. Set up a proportion and solve for x. Show all working. Understanding

    1. ▵LMN ~ ▵PQR. LM = x, MN = 12, NL = 15. PQ = 8, QR = 16, RP = 20. Find x.
    2. Two similar quadrilaterals ABCD and EFGH. AB = 10, BC = x, CD = 15. EF = 6, FG = 9, GH = y. Find x and y.
    3. In the diagram, ▵XYZ ~ ▵XAB where A is on XY and B is on XZ. XA = 4, XY = 10, XB = 6. Find XZ.
    4. Two similar isosceles triangles: the first has equal sides 10 cm and base 8 cm. The second has equal sides 15 cm. Find the base of the second triangle.

  5. Two similar figures. Understanding

    Two similar trapezoids have parallel sides 6 cm and 10 cm (first trapezoid) and 9 cm and 15 cm (second trapezoid). The non-parallel sides of the first are 5 cm and 5 cm.

    1. Find the scale factor from the first trapezoid to the second.
    2. Find the non-parallel sides of the second trapezoid.
    3. Find the perimeter of each trapezoid.
    4. The area of the first trapezoid is 32 cm². Find the area of the second trapezoid.
    5. Verify the area ratio equals k².

  6. A vertical flagpole casts a shadow of 15 m at the same time that a 1.8 m fence post casts a shadow of 2.5 m. Understanding

    1. Draw a diagram showing the two similar right triangles formed.
    2. Identify the corresponding sides.
    3. Set up and solve a proportion to find the height of the flagpole.
    4. What assumption must you make about the sun’s rays for the triangles to be similar?

  7. A model aeroplane is built to a scale of 1 : 72. Problem Solving

    1. The model has a wingspan of 43.5 cm. Find the real aeroplane’s wingspan in metres.
    2. The real aeroplane is 58.32 m long. Find the length of the model in centimetres.
    3. The model’s cockpit window has an area of 2.4 cm². Find the area of the real cockpit window in cm², then convert to m².
    4. A section of the model has a volume of 15 cm³. Find the corresponding real volume in m³.

  8. In the figure, DE is parallel to BC. AD = 5 cm, DB = 3 cm, DE = 6 cm, and AE = 4 cm. Problem Solving

    1. Explain why ▵ADE ~ ▵ABC.
    2. Find the scale factor from ▵ADE to ▵ABC.
    3. Find BC.
    4. Find EC.
    5. Find the ratio of the areas of ▵ADE and ▵ABC.

  9. A photograph 12 cm wide and 18 cm tall is to be enlarged to fit a frame 30 cm wide. Problem Solving

    1. Find the scale factor of the enlargement.
    2. Find the new height of the photograph after enlargement.
    3. Find the area of the original photograph.
    4. Find the area of the enlarged photograph.
    5. A border of 2 cm is added around the enlarged photo. Find the total dimensions and area of the framed photo (including the border).

  10. Two similar rectangles: the larger has area 180 cm² and length 15 cm. The scale factor from smaller to larger is 2.5. Problem Solving

    1. Find the width of the larger rectangle.
    2. Find the scale factor from larger to smaller.
    3. Find the dimensions of the smaller rectangle.
    4. Find the area of the smaller rectangle. Verify using the area scale factor.
    5. Find the perimeters of both rectangles. Verify the perimeter ratio equals k.
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