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Geometry - Similarity and congruence

Grade 11IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Congruence: Two shapes are congruent if they are identical in size and shape. All corresponding sides and angles are equal.

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Triangle Congruence Criteria: SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), and RHS (Right-angle, Hypotenuse, Side).

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Similarity: Two shapes are similar if they have the same shape but different sizes. Corresponding angles are equal, and corresponding sides are in the same proportion.

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Triangle Similarity Criteria: AA (two angles equal), SSS (all three sides proportional), and SAS (two sides proportional and the included angle equal).

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Linear Scale Factor (k): The ratio of any two corresponding lengths in similar figures.

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Area and Volume Scale Factors: If the linear scale factor is k, the area scale factor is k2k^2 and the volume scale factor is k3k^3.

📐Formulae

Linear Scale Factor: k=Length2Length1k = \frac{\text{Length}_2}{\text{Length}_1}

Area Ratio: Area2Area1=k2=(l2l1)2\frac{\text{Area}_2}{\text{Area}_1} = k^2 = \left(\frac{l_2}{l_1}\right)^2

Volume Ratio: Volume2Volume1=k3=(l2l1)3\frac{\text{Volume}_2}{\text{Volume}_1} = k^3 = \left(\frac{l_2}{l_1}\right)^3

Side Proportionality: aA=bB=cC\frac{a}{A} = \frac{b}{B} = \frac{c}{C}

💡Examples

Problem 1:

Two mathematically similar cylinders have heights of 4 cm and 12 cm. If the smaller cylinder has a surface area of 50 cm250\text{ cm}^2, find the surface area of the larger cylinder.

Solution:

450 cm2450\text{ cm}^2

Explanation:

First, find the linear scale factor k=124=3k = \frac{12}{4} = 3. Since we are dealing with area, use the area scale factor k2=32=9k^2 = 3^2 = 9. The area of the larger cylinder is 50×9=450 cm250 \times 9 = 450\text{ cm}^2.

Problem 2:

Triangle ABC is similar to Triangle DEF. AB = 5 cm and DE = 15 cm. If the volume of a prism with cross-section ABC is 20 cm320\text{ cm}^3, what is the volume of a similar prism with cross-section DEF?

Solution:

540 cm3540\text{ cm}^3

Explanation:

The linear scale factor k=155=3k = \frac{15}{5} = 3. For volume, the scale factor is k3=33=27k^3 = 3^3 = 27. The volume of the larger prism is 20×27=540 cm320 \times 27 = 540\text{ cm}^3.

Problem 3:

In triangle ABC, a line XY is drawn parallel to BC such that X is on AB and Y is on AC. If AX = 3 cm, XB = 6 cm, and XY = 4 cm, calculate the length of BC.

Solution:

12 cm12\text{ cm}

Explanation:

Triangles AXY and ABC are similar because XY is parallel to BC (corresponding angles are equal). The length of AB is AX+XB=3+6=9 cmAX + XB = 3 + 6 = 9\text{ cm}. The linear scale factor k=ABAX=93=3k = \frac{AB}{AX} = \frac{9}{3} = 3. Therefore, BC=XY×3=4×3=12 cmBC = XY \times 3 = 4 \times 3 = 12\text{ cm}.