The same ratio in three costumes
Recognising one similar-triangle ratio behind a shadow, a mast and a scale drawing.
Scenario
At a fixed moment one afternoon, a person 1.8 m tall casts a shadow 2.4 m long on level ground.
Everything in this set happens at that same moment and on that same ground, except where a question says otherwise.
Answers in order, explanations withheld, and a report on where the chain broke.
- 23.1Similar trianglesFoundation
A tree nearby casts a shadow 20 m long. How tall is the tree?
- A15 m
- B26.7 m
- C12 m
- D18 m
Show the answer and reasoningHide the reasoning
15 m
The sun's angle is the same for both, so the triangles are similar and height over shadow is constant: 1.8 / 2.4 = 0.75. The tree is 0.75 x 20 = 15 m. Inverting the ratio gives 26.7, which would make the tree taller than its shadow when the person is not.
- 23.2Similar trianglesStandard
Builds on: Question 1 fixed the ratio at 0.75. This is the identical relationship read in the opposite direction, which is where the reciprocal errors live.
A mast is 12 m tall. How long is its shadow at the same moment?
- A9 m
- B16 m
- C14.4 m
- D10 m
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16 m
The same ratio, used the other way: shadow is height divided by 0.75, so 12 / 0.75 = 16 m. Multiplying by 0.75 gives 9, which is the answer to "how tall is a 12 m shadow's object".
- 23.3Similar trianglesStandard
Builds on: The same proportional reasoning as questions 1 and 2, with the ratio supplied rather than measured. Recognising it as the same tool is the point of putting it here.
On a plan drawn to a scale of 1 : 50, a wall measures 6 cm. How long is the real wall?
- A3 m
- B30 m
- C300 m
- D0.12 m
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3 m
Each centimetre on the plan stands for 50 cm, so 6 cm stands for 6 x 50 = 300 cm, which is 3 m. The choice 300 m is that figure with the centimetres never converted, and 0.12 m divides by the scale instead of multiplying.
- 23.4Similar trianglesChallenge
Builds on: Questions 1 to 3 all used the ratio successfully, which is what makes this question worth asking: three successes in a row are exactly what persuades a reader that a ratio travels further than it does.
Which of these could the 0.75 ratio NOT be used to find?
- AThe height of a building from its shadow, an hour later the same day.
- BThe height of a flagpole from its shadow at the same moment.
- CThe length of a wall's shadow at the same moment, given the wall's height.
- DThe height of a statue from its shadow at the same moment.
Show the answer and reasoningHide the reasoning
The height of a building from its shadow, an hour later the same day.
The ratio is a fact about the sun's angle at one moment, not about the ground or the objects. An hour later the sun has moved and the ratio has changed, so the same arithmetic gives a confidently wrong height. Everything else on the list is at the stated moment.