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Question

There is a geometric theorem that says "If two lines in a plane are perpendicular to the same line, they are parallel to each other." Explain why this is true by writing and comparing equations for two different lines that are perpendicular to .

Answer 1

I apologize for the formatting; this is the first time I'm writing a response, and I am still getting used to these comment boxes.

Take two lines which have different y-intercepts, say (0, 2) and (0, 5). (Note: You can use any two points on the y-axis, so long as each has an x-coordinate of 0.) Since each line is perpendicular to the line y=-(1/3)x, the slope of each line will be the negative inverse of the slope of the line to which they are perpendicular.

This means the slope of the first line is -[-1/(1/3)]=+3. That means the first line has an equation of y=3x+2. This also means that the slope of the second line is -[-1/(1/3)]=+3. That means the second line has an equation of y=3x+5.

Since both lines are different (intercept of 2 vs. intercept of 5), and the both have the same slope (3), they will never intersect. Confirm this by setting the equations equal to each other, 3x+2=3x+5. Subtracting 2 from each side gives 3x=3x+3. Subtract 3x from each side gives us 0=3. Therefor, they intersect when 0=3. Since this statement is always false, these lines will never intersect. Therefor, they are parallel from the definition of parallel lines.

Answer 2

Perpendicular lines have opposite reciprocal slopes. Therefore in the equation y = -1/3x the perpendicular lines would have a slope of 3. m = -1/3 perpendicular slope would be the opposite reciprocal m = 3/1 or 3.

So any equation with the slope of 3 would be perpendicular.

ex. y = 3x -1 would be perpendicular to y = -1/3x

y = 3x + 2 would also be perpendicular to y = -1/3x

But since y = 3x - 1 and y = 3x + 2 have the same slopes (m) they are parallel.