How long is the pole's shadow?
A hard-level "Length" problem.The sun casts parallel rays; a lamp spreads from a point. Both use similar triangles, but the big triangle sits in a different place.
GivenA pole 2 m tall stands 6 m from a lamp 5 m high.
Hints
- Unlike sunlight, a lamp spreads its light from a single point. So the tip of the shadow sits where the lamp, the top of the pole and the shadow's tip line up.
- Look at the big triangle: lamp, foot of the lamp, tip of the shadow. Its height is the lamp's height and its base runs from the lamp's foot to the shadow's tip.
- Inside it sits a smaller triangle — top of the pole, foot of the pole, tip of the shadow. Both have a vertical side and share the angle at the bottom right, so they are the same shape.
- The vertical sides are in the ratio 5 : 2, and since the shapes match, the horizontal sides share that ratio.
- Horizontally, lamp-to-shadow-tip is 5 parts and pole-to-shadow-tip is 2 parts. The gap of 5 − 2 = 3 parts is the 6 m between lamp and pole, so one part is 6 ÷ 3 and the shadow is 2 parts.
Answer 4m
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