When a point moves along the sides of a figure, its area or other quantities can sometimes be represented by a linear function.
Example ①
In square ABCD with side length 10 cm, point P starts at vertex A and moves toward B along side AB at 2 cm/s. Find the area y cm² of △APD x seconds after starting (0≦x≦5).
Answer: AP=2x. The base of △APD is AP=2x, and its height is AD=10. y=(1/2)×2x×10=10x
In square ABCD with side length 10 cm, point P starts at vertex A and moves toward B along side AB at 2 cm/s. Find the area y cm² of △APD x seconds after starting (0≦x≦5).
Answer: AP=2x. The base of △APD is AP=2x, and its height is AD=10. y=(1/2)×2x×10=10x
Key Points
- Express the moving point's position using x (speed×time).
- Substitute into area = (1/2)×base×height.
- Pay attention to the domain (the range in which the point remains on the side).
Practice Problems
- In triangle ABC with a base of 8 cm and a height of 6 cm, point P moves from A along the base at 2 cm/s. Express the area y cm² of △ABP in terms of x seconds (0≦x≦4).
Answer and Explanation
- Answer: y=6x. AP=2x, height=6. y=(1/2)×2x×6=6x.