When line y= 2 and parabola y= 1/2x squared intersect at points A and B as shown below solve the following problems
from Kumon level I math page 194a
1) find the points of intersection A and B
2) Find the lengths of line segments OA,OB, and AB
3) Find the area S of triangle OAB.
1) To find the points of intersection, we set y= 2 equal to y= 1/2x^2 and solve for x:
2 = 1/2x^2
4 = x^2
x = ±2
Plugging these x-values back into either equation, we find the corresponding y-values:
When x = 2, y = 1/2(2)^2 = 2
Therefore, point A is (2, 2).
When x = -2, y = 1/2(-2)^2 = 2
Therefore, point B is (-2, 2).
2) To find the lengths of line segments OA, OB, and AB, we use the distance formula:
OA = √((2-0)^2 + (2-0)^2) = √(2^2 + 2^2) = √(4+4) = √8 = 2√2
OB = √((-2-0)^2 + (2-0)^2) = √(2^2 + 2^2) = √(4+4) = √8 = 2√2
AB = √((2-(-2))^2 + (2-2)^2) = √(4^2 + 0^2) = √16 = 4
3) To find the area S of triangle OAB, we can use the formula for the area of a triangle given two side lengths and the included angle:
S = 1/2 * OA * OB * sin(θ)
Since OAB is a right triangle, sin(θ) = 1. Therefore, the area is:
S = 1/2 * 2√2 * 2√2 = 2 * 2 = 4.