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Similarity, Vectors, Transformations

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

N/A Pathway: N/A

First Principles

Objective: Master Similarity, Transformations, and Vector Geometry

Geometry is the language of shapes, scale, and movement. Whether scaling up architectural plans for the iconic Britam Tower in Nairobi or calculating the vector flight path of a cargo drone across the Great Rift Valley, understanding how shapes scale, rotate, and translate is essential.

Interactive Geometry Lab: Scale & Vector Shift

Adjust the scale factor \(k\) and translation vector \(\mathbf{t} = (t_x, t_y)\) to observe how the original triangle (blue) transforms into the image (orange).

Base length: \(40 \times 1.5 = 60\) px | Area ratio: \(k^2 = 1.5^2 = 2.25\)

(a) Similarity vs Congruence

Two geometric figures are similar if they share identical angles and their corresponding side lengths are proportional by a common scale factor \(k\). If \(k = 1\), they are congruent (identical in size and shape).

Core Analogy: Think of a smartphone camera zoom. Zooming in multiplies every linear distance on screen by \(k\), while the relative shapes and angles remain perfectly unaltered.

(b) The Area and Volume Scaling Laws

When all linear lengths scale by \(k\):

  • Perimeter and Lengths scale linearly by \(k\).
  • Surface Area scales by \(k^2\) because area is two-dimensional (length \(\times\) width).
  • Volume scales by \(k^3\) because volume is three-dimensional.

(c) Vectors as Directed Displacements

A vector \(\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}\) describes a shift of \(x\) units horizontally and \(y\) units vertically. Unlike scalar quantities (like distance or temperature), vectors encapsulate both magnitude (length) and direction.

Key Formulas

1. Length Scale Factor (\(k\))

\[ k = \frac{\text{Image Side Length}}{\text{Original Side Length}} = \frac{A'B'}{AB} \]

Corresponding sides of similar shapes are strictly proportional.

2. Area Scale Factor

\[ \text{Area}_{\text{image}} = k^{2} \times \text{Area}_{\text{original}} \]

If length doubles (\(k=2\)), area quadruples (\(k^2=4\)).

3. Volume Scale Factor

\[ \text{Volume}_{\text{image}} = k^{3} \times \text{Volume}_{\text{original}} \]

If length doubles (\(k=2\)), volume becomes 8 times larger (\(k^3=8\)).

4. Vector Magnitude & Addition

\[ |\mathbf{v}| = \sqrt{v_{x}^{2} + v_{y}^{2}} \] \[ \mathbf{u} + \mathbf{v} = \begin{pmatrix} u_x \\ u_y \end{pmatrix} + \begin{pmatrix} v_x \\ v_y \end{pmatrix} = \begin{pmatrix} u_x + v_x \\ u_y + v_y \end{pmatrix} \]

Vector addition operates component-wise; magnitude uses Pythagoras' Theorem.

5. Standard Geometric Transformations

  • Translation by \(\mathbf{t} = \begin{pmatrix} a \\ b \end{pmatrix}\): \((x, y) \rightarrow (x+a, y+b)\)
  • Enlargement by scale factor \(k\) from origin \((0,0)\): \((x, y) \rightarrow (kx, ky)\)
  • Rotation \(90^{\circ}\) clockwise about origin: \((x, y) \rightarrow (y, -x)\)
  • Rotation \(90^{\circ}\) anticlockwise about origin: \((x, y) \rightarrow (-y, x)\)
  • Rotation \(180^{\circ}\) about origin: \((x, y) \rightarrow (-x, -y)\)

Worked Examples

Example 1 (Easy): Finding Missing Lengths in Similar Triangles

Problem: Triangle \(ABC\) is similar to triangle \(DEF\). In \(ABC\), side \(AB = 6\text{ cm}\) and side \(BC = 8\text{ cm}\). The corresponding side in \(DEF\) is \(DE = 9\text{ cm}\). Find the length of side \(EF\).

  1. Find the linear scale factor (\(k\)): \[ k = \frac{DE}{AB} = \frac{9}{6} = 1.5 \]
  2. Calculate corresponding length \(EF\): \[ EF = BC \times k = 8 \times 1.5 = 12\text{ cm} \]
  3. Answer: \(EF = 12\text{ cm}\).

Example 2 (Medium): Vector Addition and Magnitude

Problem: A ship sets out from Mombasa with displacement vector \(\mathbf{a} = \begin{pmatrix} 6 \\ -2 \end{pmatrix}\) km, and then encounters a sea current displacement \(\mathbf{b} = \begin{pmatrix} -2 \\ 5 \end{pmatrix}\) km. Find the resultant position vector and its total distance from Mombasa.

  1. Compute resultant vector: \[ \mathbf{r} = \mathbf{a} + \mathbf{b} = \begin{pmatrix} 6 + (-2) \\ -2 + 5 \end{pmatrix} = \begin{pmatrix} 4 \\ 3 \end{pmatrix} \]
  2. Calculate magnitude (straight-line distance): \[ |\mathbf{r}| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5\text{ km} \]
  3. Answer: Resultant vector is \(\begin{pmatrix} 4 \\ 3 \end{pmatrix}\) and distance is \(5\text{ km}\).

Example 3 (Hard): Area Scaling & Composite Transformation

Problem: A plot of land modeled as triangle \(P\) with coordinates \((0,0)\), \((4,0)\), and \((0,6)\) has area \(12\text{ m}^2\). The plot is enlarged by a scale factor \(k = 3\) centered at the origin, followed by a translation by vector \(\mathbf{t} = \begin{pmatrix} -5 \\ 8 \end{pmatrix}\). Find the area of the transformed plot and the new coordinates of vertex \((4,0)\).

  1. Determine the transformed area: \[ \text{New Area} = \text{Original Area} \times k^2 = 12 \times 3^2 = 12 \times 9 = 108\text{ m}^2 \] Note: Translation does not alter area.
  2. Apply enlargement to vertex \((4,0)\): \[ (4, 0) \rightarrow (4 \times 3, 0 \times 3) = (12, 0) \]
  3. Apply translation \(\begin{pmatrix} -5 \\ 8 \end{pmatrix}\): \[ (12 - 5, 0 + 8) = (7, 8) \]
  4. Answer: Area is \(108\text{ m}^2\), new vertex position is \((7, 8)\).

Common Mistakes

Mistake Multiplying area by scale factor \(k\) instead of \(k^2\).
Correction Area is 2-dimensional (length \(\times\) width). When each dimension scales by \(k\), the area scales by \(k \times k = k^2\).
Why it feels right We naturally equate the scale factor directly with the final size change across all measures. Always check dimensions: 1D = \(k\), 2D = \(k^2\), 3D = \(k^3\).
Mistake Assuming parallel vectors must have identical components.
Correction Parallel vectors are scalar multiples of each other: \(\mathbf{u} = c\mathbf{v}\). For example, \(\begin{pmatrix} 2 \\ 3 \end{pmatrix}\) and \(\begin{pmatrix} 6 \\ 9 \end{pmatrix}\) are parallel because \(3 \times \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 6 \\ 9 \end{pmatrix}\).
Why it feels right Parallel lines look the same direction, but one vector can be much longer or opposite in sense to another.
Mistake Confusing Clockwise and Anticlockwise Rotation rules.
Correction A \(90^\circ\) clockwise rotation about the origin transforms \((x,y) \rightarrow (y, -x)\). An anticlockwise rotation transforms \((x,y) \rightarrow (-y, x)\).
Why it feels right It is easy to mix up which coordinate gets the negative sign. A quick sketch of \((1,2)\) turning to \((2,-1)\) confirms clockwise movement.

Real World

Transformations and vector geometry are the cornerstone of modern engineering and digital infrastructure across Africa:

1. Surveying the Konza Technopolis

Civil engineers and architects use precise 1:500 scale models to design master plans. If a rainwater drainage pond on the map has an area of \(24\text{ cm}^2\), the actual pond area on the ground is \(24 \times 500^2 = 6{,}000{,}000\text{ cm}^2 = 600\text{ m}^2\).

2. Drone Navigation & Aviation

Medical delivery drones operating in rural clinics compute their actual flight path using vector sums: \(\mathbf{v}_{\text{actual}} = \mathbf{v}_{\text{drone}} + \mathbf{v}_{\text{wind}}\). If the drone flies heading east at \(15\text{ m/s}\) \(\begin{pmatrix} 15 \\ 0 \end{pmatrix}\) and a crosswind blows north at \(8\text{ m/s}\) \(\begin{pmatrix} 0 \\ 8 \end{pmatrix}\), the drone's ground speed is \(\sqrt{15^2 + 8^2} = 17\text{ m/s}\).

3. African Art and Textile Geometry

Traditional Kitenge and Kanga fabric patterns utilize repeating transformations—translations, glide reflections, and rotations—to generate infinite intricate, symmetrical decorative motifs.

Practice

Two similar triangles have their corresponding sides in the ratio 3:5. If the perimeter of the smaller triangle is 27 cm, what is the perimeter of the larger triangle? (Type only the number, e.g., 30)
Review the concepts above.
Triangle ABC is similar to triangle DEF. The sides of triangle ABC are 6 cm, 8 cm, and 10 cm. The side of triangle DEF that corresponds to the 8 cm side of triangle ABC measures 12 cm. What is the scale factor from triangle ABC to triangle DEF? (Type only the number, e.g., 3.7)
Review the concepts above.
Vector A has components (3, 4). Vector B has components (k, 12) and is parallel to vector A. Find the value of k. (Type only the number, e.g., 42)
Review the concepts above.
Two similar triangles have their corresponding side lengths in the ratio 3:5. The perimeter of the smaller triangle is 48 cm. What is the perimeter of the larger triangle? (Type only the number, e.g., 42)
Review the concepts above.
A triangle has a base of 8 cm and a height of 5 cm. It is enlarged with a scale factor of \(\frac{3}{2}\). What is the area of the enlarged triangle in square centimetres? (Type only the number, e.g., 30)
Review the concepts above.
Triangle ABC is similar to triangle DEF. The scale factor from ABC to DEF is 3. If the area of triangle ABC is 12 cm², what is the area of triangle DEF? (Type only the number, e.g., 42)
Review the concepts above.