How can you mathematically prove that two lines are perpendicular?
To mathematically prove that two lines are perpendicular, you can use the concept of slopes of the lines.
The slope of a line is a measure of its steepness. It can be calculated using the formula: slope = (change in y-coordinate) / (change in x-coordinate).
If two lines are perpendicular, their slopes should be negative reciprocals of each other. The negative reciprocal of a number is obtained by taking the reciprocal (flipping the fraction upside down) and changing the sign (positive to negative or negative to positive).
Here are the steps to mathematically prove that two lines are perpendicular:
1. Given two lines with equations, for example:
- Line 1: y = m1x + b1
- Line 2: y = m2x + b2
2. Calculate the slopes of both lines using the given equations:
- Slope of Line 1: m1
- Slope of Line 2: m2
3. Determine the negative reciprocal of one of the slopes. Let's say you choose Line 1, so the negative reciprocal of m1 is -1/m1.
4. Compare the negative reciprocal and the slope of the other line (Line 2). If they are equal, the two lines are perpendicular.
If -1/m1 = m2, the lines are perpendicular.
If -1/m1 ≠ m2, the lines are not perpendicular.
By following these steps, you can mathematically prove whether two lines are perpendicular or not based on their slopes.