Introduction:
To find a point on the curve y=x^2 where the tangent is parallel to the chord joining (0,0) and (1,1), we need to determine the equation of the tangent line and the chord, and then find the point of intersection between them. This can be done by calculating the slopes of both the tangent line and the chord.

Calculating the slope of the chord:
The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) can be found using the formula:
m = (y₂ - y₁) / (x₂ - x₁)

In this case, the two points are (0,0) and (1,1), so the slope of the chord is:
m₁ = (1 - 0) / (1 - 0) = 1

Calculating the slope of the tangent:
The derivative of the function y = x^2 gives us the slope of the tangent line at any given point on the curve. Taking the derivative, we get:
dy/dx = 2x

Since we want the tangent line to be parallel to the chord, the slope of the tangent line must also be 1. Therefore, we can set dy/dx = 1 and solve for x to find the x-coordinate of the point where the tangent is parallel to the chord.

2x = 1
x = 1/2

Finding the y-coordinate:
To find the y-coordinate of the point, we substitute the x-coordinate (x = 1/2) into the equation of the curve y = x^2:
y = (1/2)^2
y = 1/4

Therefore, the point on the curve y = x^2 where the tangent is parallel to the chord joining (0,0) and (1,1) is (1/2, 1/4).

Summary:
1. The slope of the chord joining (0,0) and (1,1) is 1.
2. The slope of the tangent line can be found by taking the derivative of the curve y = x^2, which gives us dy/dx = 2x.
3. Setting dy/dx equal to the slope of the chord, we solve for x to find the x-coordinate of the point.
4. Substituting the x-coordinate into the equation of the curve, we find the y-coordinate of the point.
5. The point on the curve y = x^2 where the tangent is parallel to the chord is (1/2, 1/4).