Similarity and Enlargement/Reduction ③ Problems Involving Shadow Lengths
Sunlight reaches the ground in parallel rays. Therefore, the height of an object, the length of its shadow, and the rays of light form “similar” triangles with the same shape.
How to Think About Shadow Problems
- Sunlight is parallel → the object, the ground, and the rays of light form similar triangles
- At the same time, “height of the object : length of the shadow” is always the same ratio
Example 1: A person 150 cm tall has a shadow 200 cm long. At the same time, how long will the shadow of a 6 m tree be?
Solution:
① Use the same unit, centimeters → tree height 600 cm
② Proportion → 150 : 200 = 600 : x
③ x = 600 × 200 ÷ 150 = 800 cm = 8 m
Solution:
① Use the same unit, centimeters → tree height 600 cm
② Proportion → 150 : 200 = 600 : x
③ x = 600 × 200 ÷ 150 = 800 cm = 8 m
Example 2: A utility pole is 8 m tall and has a shadow 6 m long. At the same time, a building has a shadow 45 m long. How tall is the building?
Solution:
① Proportion → 8 : 6 = h : 45
② h = 8 × 45 ÷ 6 = 60 m
Solution:
① Proportion → 8 : 6 = h : 45
② h = 8 × 45 ÷ 6 = 60 m
It is because the observations are made “at the same time” that we can say the shapes are similar. If the times are different, the angle of the sun changes and the ratio no longer holds.