L4 · Similarity clinic

Learning goal

“I can run the whole similarity toolkit without prompting: scale factor from lengths, the ratio test for similarity, areas via k^2 (and back via square roots), dilations about the origin, and scale in maps and blueprints - choosing the right tool for each problem.”

Scale drawing — a drawing made with a scale factor: maps, blueprints, models and plans. Every length on it is the real length times the same k.

Scale — the scale factor written for a drawing, often as a ratio (1 : 30) or a correspondence (1 cm : 5 km, where units do the converting).

Worked example

A floor plan is drawn with k = 1/40 (plan / real). A room is 6 cm long on the plan and its floor area on the plan is 15 cm2. Question 1 (a length): real length = 6 / (1/40) = 6 x 40 = 240 cm = 2.4 m. Question 2 (an area): the area factor is (1/40)^2 = 1/1600, so the real floor area is 15 x 1600 = 24,000 cm2 = 2.4 m2. Say it plainly: the SAME k serves both questions, but lengths travel by k and areas by k^2 - naming which kind of quantity you hold is the first move of every problem in this unit.

Check yourself — explain it

The area question needed 1600 where the length question needed 40 because...

Paired problem — example still visible

With k = 2 (image / original), an original length of 9 cm and an original area of 20 cm2 become what? -> Length 18 cm; area 20 x 2^2 = 80 cm2.

Faded completion — last step is yours

"k = 0.5. A length of 30 cm becomes ______ cm. An area of 30 cm2 becomes ______ cm2." -> 15 cm; 30 x 0.25 = 7.5 cm2

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

Start practice →Take the unit quiz