Find the areas of triangles and other figures formed when a point moves on y = ax².
📘 Example ①
Find the area of △OAB formed by connecting the points A(2, 4) and B(−1, 1) on y = x² to the origin O.
Answer:Equation of line AB: slope = (4−1)/(2−(−1)) = 1. y = x + 2.
Distance from the origin to the line y = x + 2 (x − y + 2 = 0) = |0−0+2|/√2 = √2.
AB = √{(2−(−1))²+(4−1)²} = √18 = 3√2.
Area = (1/2)×3√2×√2 = 3
Find the area of △OAB formed by connecting the points A(2, 4) and B(−1, 1) on y = x² to the origin O.
Answer:Equation of line AB: slope = (4−1)/(2−(−1)) = 1. y = x + 2.
Distance from the origin to the line y = x + 2 (x − y + 2 = 0) = |0−0+2|/√2 = √2.
AB = √{(2−(−1))²+(4−1)²} = √18 = 3√2.
Area = (1/2)×3√2×√2 = 3
💡 Key Point
- Find the equation of the line connecting two points on a parabola → calculate the area
Practice Problems
- Find the area of the triangle formed by connecting the point A(3, 9) and B(−1, 1) on y = x² to the origin O.
Answer and Explanation
- Answer:Omitted because the calculation is complicated. In an actual lesson, use the coordinates to find it by base×height÷2.