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Learning Resources

Scale Drawing

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

Grade 09 Pathway: N/A

First Principles

Objective: Master the geometry and algebra of scale drawings, three-figure bearings, and angles to represent real-world spatial relationships accurately on paper.

Concrete Scenario: Imagine mapping out a school compound in Nakuru from the main gate to the science laboratory. The true ground distance is \(60\text{ m}\) along a specific direction—such as \(060^\circ\) from North. Because a \(60\text{ m}\) sheet of paper is impossible to carry, we shrink every distance by a fixed ratio called the scale factor, while keeping all angles and directions identical.

Geometric Insight: A scale drawing produces a similar figure to the actual terrain. Dilatations preserve angles exactly; therefore, a bearing measured on the ground with a magnetic compass is drawn with the exact same angle on paper using a protractor and a reference North line.

Algebraic Principles:

  • Representative Fraction (Scale): A scale of \(1 : n\) means \(1\text{ unit on paper} = n\text{ units in reality}\).
  • Scale Factor: \(s = \frac{1}{n}\). Thus, \(L_{\text{draw}} = \frac{L_{\text{real}}}{n}\) and \(L_{\text{real}} = L_{\text{draw}} \times n\). (Ensure units match before multiplying or dividing!)
  • Three-Figure Bearings: Angles measured strictly clockwise from True North, written with three digits (e.g., \(045^\circ\), \(090^\circ\), \(230^\circ\)).

Key Formulas

\[ \text{Representative Fraction (RF)} = \frac{\text{Drawing Length}}{\text{Real Length (in identical units)}} = \frac{1}{n} \]
\[ L_{\text{draw}} = \frac{L_{\text{real}}}{n} = s \times L_{\text{real}} \]
Used when translating field measurements to paper.
\[ L_{\text{real}} = L_{\text{draw}} \times n = \frac{L_{\text{draw}}}{s} \]
Used when reading actual distances off a blueprint or topographical map.
\[ \text{Three-Figure Bearing } \theta \in [000^\circ, 360^\circ) \]
Measured in a clockwise direction starting from True North (\(000^\circ\)).
\[ \text{Back Bearing} = \begin{cases} \theta + 180^\circ & \text{if } \theta < 180^\circ \\ \theta - 180^\circ & \text{if } \theta \ge 180^\circ \end{cases} \]
The direction from the destination back to the starting point.
\[ \text{Area}_{\text{real}} = \text{Area}_{\text{draw}} \times n^2 \]
Area scales with the square of the linear scale factor.

Worked Examples

Example 1 (Easy — Ground to Paper Conversion):

Problem: A surveyor measures a boundary wall of length \(85\text{ m}\) on a farm. Determine the length of this wall on a blueprint with a scale of \(1 : 500\).

  1. Convert units to centimetres: \[ L_{\text{real}} = 85\text{ m} = 85 \times 100\text{ cm} = 8500\text{ cm} \]
  2. Apply the scale ratio \(1 : 500\): \[ L_{\text{draw}} = \frac{8500\text{ cm}}{500} = 17\text{ cm} \]
  3. Conclusion: The line drawn on the map is \(17\text{ cm}\).
Example 2 (Medium — Interpreting Map Distance & Back Bearing):

Problem: On a topographical map with scale \(1 : 25\,000\), the vector from Matatu Terminal \(A\) to Market \(B\) measures \(4.8\text{ cm}\) on a bearing of \(075^\circ\). Calculate the real ground distance in kilometres, and state the bearing of \(A\) from \(B\) (the back bearing).

  1. Calculate real ground distance: \[ L_{\text{real}} = 4.8\text{ cm} \times 25\,000 = 120\,000\text{ cm} \]
  2. Convert to kilometres: \[ 120\,000\text{ cm} = \frac{120\,000}{100}\text{ m} = 1200\text{ m} = 1.2\text{ km} \]
  3. Find the back bearing (direction of \(A\) from \(B\)): Since the forward bearing is \(\theta = 075^\circ < 180^\circ\): \[ \text{Back Bearing} = 075^\circ + 180^\circ = 255^\circ \]
  4. Conclusion: The actual distance is \(1.2\text{ km}\) and the bearing of \(A\) from \(B\) is \(255^\circ\).
Example 3 (Hard — Two-Leg Traverse with Scale and Area):

Problem: A triangular piece of land \(PQR\) is surveyed. From \(P\), a boundary runs due North (\(000^\circ\)) for \(60\text{ m}\) to \(Q\). From \(Q\), another boundary runs due East (\(090^\circ\)) for \(80\text{ m}\) to \(R\). Using a scale of \(1 : 1000\), determine the drawing lengths \(PQ\) and \(QR\), the length of \(PR\) on paper, and the actual land area in square metres.

  1. Calculate drawing lengths: \[ PQ_{\text{draw}} = \frac{60 \times 100\text{ cm}}{1000} = 6\text{ cm} \] \[ QR_{\text{draw}} = \frac{80 \times 100\text{ cm}}{1000} = 8\text{ cm} \]
  2. Find map hypotenuse \(PR_{\text{draw}}\) using Pythagoras: Since North and East are perpendicular (\(\angle PQR = 90^\circ\)): \[ PR_{\text{draw}} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\text{ cm} \]
  3. Calculate the actual area: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 80\text{ m} \times 60\text{ m} = 2400\text{ m}^2 \]

Common Mistakes

Mistake 1: Measuring Bearings Anticlockwise or from the Horizontal Axis.
Why it feels right: In Cartesian coordinate geometry, standard angles are measured anticlockwise from the positive \(x\)-axis (East).
Correction: In navigation and surveying, bearings are ALWAYS measured clockwise starting from True North (\(000^\circ\)). An angle of \(60^\circ\) East of North is written as \(060^\circ\).
Mistake 2: Mixing Units during Scale Calculation.
Why it feels right: Learners often divide metres directly by the scale factor (e.g., dividing \(120\text{ m}\) by \(500\) to get \(0.24\)) without specifying units, leading to confusion between metres, centimetres, and millimetres.
Correction: Always convert the real-world distance to centimetres (multiply by \(100\)) before dividing by the scale denominator \(n\).
Mistake 3: Scaling Areas with the Linear Scale Factor \(n\) instead of \(n^2\).
Why it feels right: If lengths scale by \(1:500\), it is tempting to assume area also scales by \(500\).
Correction: Area has two dimensions (length \(\times\) width). Therefore, area scales by the square of the linear scale factor: \(\text{Area}_{\text{real}} = \text{Area}_{\text{draw}} \times n^2\).

Real World

Cadastral Surveying & Land Title Deeds in Kenya: Ministry of Lands surveyors produce deed plans (R.I.M. - Registry Index Maps) using scales such as \(1:2500\) or \(1:5000\). Precise bearings delineate shamba boundaries to prevent land disputes.
Civil Engineering & SGR Rail Design: Engineers laying out the Standard Gauge Railway (SGR) track alignment use precise bearings and curve radius scale drawings to ensure high-speed trains navigate terrain safely.
Aviation & Maritime Navigation: Pilots departing from Jomo Kenyatta International Airport (JKIA) receive departure routes given in three-figure compass headings (bearings) relative to magnetic and true north.
Architectural Floor Plans: Architects draw residential buildings using standard metric scales (e.g., \(1:50\) or \(1:100\)) so that masons and contractors can accurately construct full-size rooms on site.

Practice

Jackson is drawing a scale map of his shamba. He uses a scale where 1 cm on paper represents 5 metres in reality. If the actual fence around the shamba is 120 metres long, how many centimetres long should the fence be on his drawing? (Type only the number, e.g., 42)
Review the concepts above.
On a physical map, a scale is given as 1 cm to represent 8 metres on the ground. If a straight access road is drawn as 3.5 cm on the map, what is the actual length of the road in metres? (Type only the number, e.g., 42)
Review the concepts above.
A road construction blueprint uses a scale ratio of 1 : 2500. If a section of highway is drawn as 6 cm long on the blueprint, what is the actual length of that highway section in metres? (Type only the number, e.g., 42)
Review the concepts above.
An architect is building a model of a commercial SACCO building that stands 15 metres tall. If the model is constructed at a scale of 1 : 50, how tall will the model be in centimetres? (Type only the number, e.g., 42)
Review the concepts above.
Jackson draws a map of a rectangular garden plot using a scale of 1 : 50. If the real perimeter of the fence around the garden is 30 metres, what is the total perimeter of the fence as drawn on the map, in centimetres? (Type only the number, e.g., 42)
Review the concepts above.
A surveyor drafts a scale plan of a triangular plot using a scale of 1 cm = 5 m (which is 1 : 500). On the drawing, the triangle has a base of 9 cm and a perpendicular height of 4 cm. What is the actual area of the plot in square metres? (Type only the number, e.g., 42)
Review the concepts above.