This lesson is about creating accurate, scaled-down drawings to find unknown heights and distances in real-world scenarios. It primarily deals with problems in a vertical plane (measuring up and down) using angles of elevation and depression. This is a practical skill where neatness and a methodical approach are key.

1. Core Concepts & Vocabulary

• Scale: The ratio that links the drawing length to the actual length. Example: 1 cm : 5 m means 1 cm on your paper represents 5 metres in reality.

• Angle of Elevation: The angle measured UPWARDS from the horizontal (eye-level) to an object above.

• Angle of Depression: The angle measured DOWNWARDS from the horizontal (eye-level) to an object below.

• Key Geometric Fact: The angle of depression from point A to point B is equal to the angle of elevation from point B to point A (they form alternate "Z-angles").

2. The 5-Step Method for Solving Scale Diagram Problems

Follow these steps for every question. This structured approach prevents mistakes.

1. DRAW A ROUGH SKETCH: Before you measure anything, draw a quick, freehand diagram of the situation. Label it with all the heights, distances, and angles given in the problem. This is your plan.

2. CHOOSE AND STATE YOUR SCALE: Pick a simple scale that lets the drawing fit neatly on your page. Write it down clearly at the top of your answer (e.g., Scale: 1 cm represents 2 m).

3. CONSTRUCT THE ACCURATE DRAWING: Use a ruler and protractor to draw the diagram precisely.

   • Start with a baseline (usually the horizontal ground).

   • Use the protractor to measure all angles from the horizontal line.

   • Use the ruler to draw lines to the correct scaled length.

4. MEASURE THE REQUIRED LENGTH: Use your ruler to measure the line on your drawing that represents the unknown height or distance the question is asking for.

5. CONVERT BACK TO THE REAL-WORLD LENGTH: Use your scale to calculate the actual length. State your final answer clearly with the correct units.

3. A Detailed Worked Example

Question: A person stands 12 metres away from the base of a flagpole. The angle of elevation to the top of the pole is 40°. Find the height of the flagpole using a scale diagram.

Solution using the 5-Step Method:

1. Rough Sketch:

   • Draw a right-angled triangle.

   • Label the base "12 m".

   • Label the angle at the person's position "40°".

   • Label the vertical side "Height?".

2. Choose Scale: The distance is 12 m. A good scale would be 1 cm : 2 m.

   • (This means the 12 m base will be 12 / 2 = 6 cm long in our drawing).

3. Construct:

   • Draw a horizontal line segment AB = 6 cm long.

   • Place the protractor at point A and measure a 40° angle. Draw a line from A through this angle.

   • Draw a perpendicular line upwards from B until it intersects the line from A. Label the intersection point C. ΔABC is your scale drawing.

4. Measure:

   • Use a ruler to measure the length of the vertical line segment BC on your drawing.

   • You should measure approximately 5 cm.

5. Convert:

   • Use the scale (1 cm : 2 m) to find the actual height.

   • Actual Height = (Length on drawing) × (Scale factor)

   • Actual Height = 5 cm × 2 m/cm = 10 m.

   • Final Answer: The height of the flagpole is approximately 10 m.

4. Important Points to Remember

• Leave construction lines visible. Don't erase them.

• Observer's Height: If the problem mentions the height of the observer, your horizontal "eye-level" line must be drawn that far above the ground line.

• Accuracy is crucial. Use a sharp pencil and be precise with your ruler and protractor.

• Always finish with a concluding sentence stating the final, real-world answer with its units.