Convert to common units before applying a length scale.
Core & Extended · Syllabus C4.2, C4.3, E4.2, E4.3
Notes and key methods
A scale 1:N means one drawing unit represents N of the same real units. Convert centimetres and metres consistently before multiplying or dividing. Label an accurate scale drawing with its scale and units.
For an SSS triangle construction, draw one side with a ruler, then draw arcs centred at its endpoints using the other two side lengths as radii. Their intersection fixes the third vertex. Retain the arcs as evidence, check the triangle inequality first, and do not replace an accurate construction with a freehand sketch.
Check the conditions
Use the stated units and precision.
Show the method as well as the final answer.
Three worked examples
Example 1 · Scale drawings and accurate constructions
A scale drawing uses 1:200. All construction lengths in the questions are actual drawing lengths.
A drawing segment is 9 cm. Find its real length in metres.
A real segment is \(14\) m. Find its drawing length in cm.
Describe an accurate ruler-and-compass construction of the stated SSS triangle.
Using side 8 cm as the base, what is the radius in cm of the arc centred at its first endpoint if the side from that endpoint to the third vertex is 6 cm?
Show the reasoning
(a)
Apply the length scale and convert centimetres to metres.
Multiply 9 by 200 to get real centimetres, then divide by 100: \(18\) m.
Answer: \(18\).
(b)
Convert to common units and divide by the scale denominator.
Convert the real length to 1400 cm, then divide by 200 to obtain 7 cm.
Answer: \(7\).
(c)
The three lengths satisfy the triangle inequality. Each arc fixes a distance from one endpoint, so their intersection meets both remaining side requirements. Either reflected intersection gives a congruent triangle.
Answer: Draw the base 8 cm; intersect arcs of radii 6 cm and 7 cm from its respective endpoints; join the intersection to both endpoints.
(d)
Match each construction arc to the side connecting its centre and the new vertex.
That arc must have radius 6 cm to impose the stated distance from its centre.
Answer: \(6\).
Example 2 · Scale drawings and accurate constructions
A scale drawing uses 1:300. All construction lengths in the questions are actual drawing lengths.
A drawing segment is 3 cm. Find its real length in metres.
A real segment is \(24\) m. Find its drawing length in cm.
Describe an accurate ruler-and-compass construction of the stated SSS triangle.
Using side 9 cm as the base, what is the radius in cm of the arc centred at its first endpoint if the side from that endpoint to the third vertex is 7 cm?
Show the reasoning
(a)
Apply the length scale and convert centimetres to metres.
Multiply 3 by 300 to get real centimetres, then divide by 100: \(9\) m.
Answer: \(9\).
(b)
Convert to common units and divide by the scale denominator.
Convert the real length to 2400 cm, then divide by 300 to obtain 8 cm.
Answer: \(8\).
(c)
The three lengths satisfy the triangle inequality. Each arc fixes a distance from one endpoint, so their intersection meets both remaining side requirements. Either reflected intersection gives a congruent triangle.
Answer: Draw the base 9 cm; intersect arcs of radii 7 cm and 8 cm from its respective endpoints; join the intersection to both endpoints.
(d)
Match each construction arc to the side connecting its centre and the new vertex.
That arc must have radius 7 cm to impose the stated distance from its centre.
Answer: \(7\).
Example 3 · Scale drawings and accurate constructions
A scale drawing uses 1:400. All construction lengths in the questions are actual drawing lengths.
A drawing segment is 4 cm. Find its real length in metres.
A real segment is \(36\) m. Find its drawing length in cm.
Describe an accurate ruler-and-compass construction of the stated SSS triangle.
Using side 10 cm as the base, what is the radius in cm of the arc centred at its first endpoint if the side from that endpoint to the third vertex is 8 cm?
Show the reasoning
(a)
Apply the length scale and convert centimetres to metres.
Multiply 4 by 400 to get real centimetres, then divide by 100: \(16\) m.
Answer: \(16\).
(b)
Convert to common units and divide by the scale denominator.
Convert the real length to 3600 cm, then divide by 400 to obtain 9 cm.
Answer: \(9\).
(c)
The three lengths satisfy the triangle inequality. Each arc fixes a distance from one endpoint, so their intersection meets both remaining side requirements. Either reflected intersection gives a congruent triangle.
Answer: Draw the base 10 cm; intersect arcs of radii 8 cm and 9 cm from its respective endpoints; join the intersection to both endpoints.
(d)
Match each construction arc to the side connecting its centre and the new vertex.
That arc must have radius 8 cm to impose the stated distance from its centre.
Answer: \(8\).
Common mistakes
Keep intermediate values exact until the final step.