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Congruence, Similarity, Pythagoras

Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.

Form 2 Pathway: N/A

First Principles

Objective

Identify congruent and similar shapes, and apply Pythagoras' theorem to solve geometric problems.

Concrete Scenario: A matatu driver in Nairobi needs a ramp to load heavy cargo onto a lorry. The vertical height of the lorry bed is \(3\,\text{m}\) and the ground distance is \(4\,\text{m}\). What length of wooden timber must be cut for the ramp? Using Pythagoras' theorem (\(a^2 + b^2 = c^2\)), the driver calculates \(\sqrt{3^2 + 4^2} = 5\,\text{m}\).

Geometric Insight:

  • Congruence (\(\cong\)): Identical in both shape and size. Corresponding angles and corresponding side lengths are exactly equal.
  • Similarity (\(\sim\)): Same shape, but different sizes. Corresponding angles are equal, while corresponding sides are in the same constant ratio \(k\) (scale factor).
  • Pythagoras' Theorem: In any right-angled triangle, the area of the square on the hypotenuse equals the sum of the areas of the squares on the other two sides.

Visualizing Pythagoras (3-4-5 Triangle)

3² = 94² = 16c² = 25b = 4a = 3

\(a^2 + b^2 = c^2 \implies 9 + 16 = 25 \implies c = 5\)

Key Formulas

\[a^2 + b^2 = c^2\] — Pythagorean Theorem: Applies exclusively to right-angled triangles where \(c\) is the hypotenuse.
\[c = \sqrt{a^2 + b^2},\quad a = \sqrt{c^2 - b^2},\quad b = \sqrt{c^2 - a^2}\] — Rearranged forms to calculate missing legs or hypotenuse.
\[\text{SSS},\; \text{SAS},\; \text{ASA},\; \text{RHS}\] — Congruence Criteria: Conditions under which two triangles are identical in size and shape.
\[\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k\] — Similarity Ratio: Corresponding side lengths of similar triangles are in constant proportion \(k\).
\[\text{Area Ratio} = k^2\] — The ratio of areas of similar figures equals the square of the linear scale factor.

Worked Examples

Example 1 (Easy) — Testing Congruence (SSS):

Problem: Triangle A has sides 5 cm, 7 cm, and 9 cm. Triangle B has sides 9 cm, 5 cm, and 7 cm. Are they congruent?

  1. Compare corresponding side lengths: both triangles have side lengths of 5 cm, 7 cm, and 9 cm.
  2. Apply the SSS (Side-Side-Side) rule: Since all three corresponding sides are equal, the two triangles are congruent.
Example 2 (Medium) — Similar Triangles Scale Factor:

Problem: Triangles ABC and DEF are similar (\(\triangle ABC \sim \triangle DEF\)). If \(AB = 8\,\text{cm}\), \(DE = 12\,\text{cm}\), and \(BC = 6\,\text{cm}\), find \(EF\).

  1. Calculate linear scale factor \(k = \frac{DE}{AB} = \frac{12}{8} = 1.5\).
  2. Multiply corresponding side \(BC\) by \(k\): \[EF = BC \times k = 6 \times 1.5 = 9\,\text{cm}\]
Example 3 (Hard) — Altitude to Hypotenuse:

Problem: In a right-angled triangle with legs \(a = 6\,\text{cm}\) and \(b = 8\,\text{cm}\), calculate the length of the altitude \(h\) drawn from the right angle to the hypotenuse.

  1. Calculate the hypotenuse \(c\): \[c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = 10\,\text{cm}\]
  2. Express the area of the triangle in two ways: \[\text{Area} = \frac{1}{2} \times a \times b = \frac{1}{2} \times 6 \times 8 = 24\,\text{cm}^2\] \[\text{Area} = \frac{1}{2} \times c \times h = \frac{1}{2} \times 10 \times h = 5h\]
  3. Equate the areas and solve for \(h\): \[5h = 24 \implies h = 4.8\,\text{cm}\]

Common Mistakes

Mistake Applying Pythagoras' theorem (\(a^2 + b^2 = c^2\)) to non-right-angled triangles.
Correction Pythagoras' theorem applies only if one angle is exactly 90°. For non-right triangles, use the Cosine Rule.
Why it feels right The formula is familiar, so students tend to apply it to any given triangle without checking for the right angle.
Mistake Assuming similar triangles are congruent.
Correction Similar triangles have equal angles but scaled side lengths. Congruent triangles must have both equal angles AND equal side lengths.
Why it feels right Because similar triangles look identical in shape, it is easy to forget that their physical dimensions differ.

Real World

Construction & Carpentry: Builders in Kenya use the 3-4-5 Pythagorean triple rule to ensure wall corners form perfect 90° right angles before laying foundations.
Roof Truss Design: Fabricating matching roof trusses requires verifying SSS congruence so the roof pitch is uniform across the entire structure.
Architectural Models: Creating scale models of buildings using similarity scale factors (e.g., 1:50) to project physical dimensions accurately.

Practice

A matatu driver builds a right-angled triangular ramp to load goods. The horizontal distance along the ground is 4 m and the ramp length (hypotenuse) is 5 m. What is the vertical height of the ramp in metres? (Type only the number, e.g., 3)
Review the concepts above.
In a right-angled triangle, one leg measures 9 cm and the hypotenuse measures 15 cm. Find the length of the other leg in centimetres. (Type only the number, e.g., 12)
Review the concepts above.
A right-angled triangle has legs of lengths 7 cm and 24 cm. What is the length of the hypotenuse in centimetres? (Type only the number, e.g., 25)
Review the concepts above.
Triangles PQR and STU are similar (\(\triangle PQR \sim \triangle STU\)). If PQ = 5 cm, QR = 12 cm, and ST = 15 cm, what is the length of TU in centimetres? (Type only the number, e.g., 36)
Review the concepts above.
Two similar 3-4-5 right-angled triangles have their short legs in the ratio 3:5. If the short leg of the smaller triangle measures 9 cm, what is the hypotenuse of the larger triangle in centimetres? (Type only the number, e.g., 25)
Review the concepts above.
In a right-angled triangle with legs of 6 cm and 8 cm, an altitude is drawn from the right angle to the hypotenuse. What is the length of this altitude in centimetres? (Type only the number, e.g., 4.8)
Review the concepts above.