Answer:
- B = (-2, 2), A = (2, 2)
- OA = 2√2, OB = 2√2, AB = 4
- area = 4 square units
Explanation:
You want the points of intersection of the line y=2 with the parabola y=1/2x², the lengths of segments between those points and the vertex, and the area of the triangle formed.
1. Intersection
The points of intersection will satisfy the two equations ...
Setting the y-expressions equal, we have ...
2 = 1/2x²
4 = x² . . . . . . multiply by 2
±2 = x . . . . . . . take the square root
The y-values of each point are 2, so the two points of intersection are ...
B = (-2, 2)
A = (2, 2)
2. Lengths
We recognize that segments OA and OB are each the hypotenuse of a right triangle that is 2 units on each side. The Pythagorean theorem tells you that hypotenuse has length ...
OA = OB = √(2² +2²) = √(2²·2)
OA = OB = 2√2
Of course AB is 4 units, the difference between the x-values of its ends.
3. Area
The area of the triangle is ...
A = 1/2bh
where the base b is AB = 4 units, and the height is y=2.
A = 1/2·4·2 = 4 . . . . square units
The area of the triangle is 4 square units.