Example Card: Writing Equations of Perpendicular Lines
Write equations of perpendicular lines using negative reciprocal slopes in Grade 9. Practice slope-intercept form for lines through a given point.
Key Concepts
Ready to make a sharp 90 degree turn? This example shows how to build a line that's perfectly perpendicular to another, focusing on the second key idea.
Example Problem Write an equation in slope intercept form for the line that passes through (4, 1) and is perpendicular to a line with equation $y = 2x + 5$.
Step by Step 1. First, find the slope of the given line, $y = 2x + 5$. The slope is $ 2$. 2. A perpendicular line has a slope that is the negative reciprocal of the original slope. The negative reciprocal of $ 2$ is $\frac{1}{2}$. So, for our new line, $m = \frac{1}{2}$. 3. Use the point slope formula, $y y 1 = m(x x 1)$, with the new slope and the given point (4, 1). 4. Substitute the values into the formula. $$ y ( 1) = \frac{1}{2}(x 4) $$ 5. Simplify the equation. $$ y + 1 = \frac{1}{2}(x 4) $$ 6. Apply the Distributive Property to the right side. $$ y + 1 = \frac{1}{2}x 2 $$ 7. Isolate $y$ by subtracting $1$ from both sides to get the slope intercept form. $$ y = \frac{1}{2}x 3 $$.
Common Questions
How do you find the slope of a perpendicular line?
Take the negative reciprocal of the original slope. If the given line has slope 2/3, the perpendicular line has slope -3/2. Their product equals -1.
How do you write the equation of a perpendicular line through a point?
Find the negative reciprocal slope, then use point-slope form: y-y1 = m(x-x1). Substitute the point and new slope, then convert to slope-intercept form.
What makes two lines perpendicular?
Two lines are perpendicular when their slopes are negative reciprocals (m1 times m2 = -1), forming a 90-degree angle at their intersection.