Write the equation of a line that hits the origin at a 45-degree angle. Explain how you determined your answer.

In order to write the equation of a line, we need to know either two points that belong to that line or one point and the line's slope. In this case, since the line "hits the origin" at a 45-degree angle, this means that:

1) The line passes through the...

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In order to write the equation of a line, we need to know either two points that belong to that line or one point and the line's slope. In this case, since the line "hits the origin" at a 45-degree angle, this means that:

1) The line passes through the origin. In other words, point (0,0) belongs to the line.

2) The line makes a 45-degree angle with the x-axis. This information helps determine the slope of the line.

The slope can be calculated as "rise over run," or the change in the vertical component divided by the corresponding change in the horizontal component:

`m=(Delta y)/(Delta x)` .

Consider a right triangle created by the line, the horizontal segment with the length of

`Delta x` and the vertical segment with the length `Delta y` . The angle that the line makes with the horizontal (x-axis) lies opposite the vertical segment. Thus, the tangent of this angle, which by definition equals the length of the opposite side divided by the length of the adjacent side, is the same thing as the slope of the line: `(Delta y)/(Delta x)` .

If the slope of the line is the tangent of the angle the line makes with the x-axis, then the slope of the line that makes a 45-degree angle with the x-axis is tan(45) = 1.

So, the line we are looking for passes through the point (0,0) and has the slope of m = 1.

Using point-slope form, the equation of the line can be written as

y - 0 = 1(x - 0), or simply

y = x.

The equation of the line that hits the origin at a 45-degree angle is y = x.

The graph of this line is below. Note that this line bisects the angle between the coordinate axes.

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