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High school geometry

Course: High school geometry > Unit 6

Lesson 5: Equations of parallel & perpendicular lines
Math
High school geometry
Analytic geometry
Equations of parallel & perpendicular lines
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Analytic geometry FAQ

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Frequently asked questions about analytic geometry

How do we divide a line segment into two segments with a certain ratio of lengths?

There a few ways to solve this type of problem. One way to is to apply the given ratio to both the horizontal and vertical displacements between the points.
For example, if point A is at left parenthesis, 0, comma, 0, right parenthesis, point C is at left parenthesis, 8, comma, 4, right parenthesis, and we want to find point B such that A, B is start fraction, 3, divided by, 4, end fraction of A, C, we can use a strategy like this:
A line segment with a positive slope containing point A located at zero, zero, point B located at six, three, and point C located at eight, four. A horizontal dotted line connects point B to a vertical tape diagram of four one unit parts to the right of the line. A vertical dotted line connects point B to a horizontal tape diagram of four two unit parts below the line. The dotted lines show that point B has a displacement of one out of four vertical parts down and one out of four horizontal parts to the right.
  • The horizontal displacement is 8 units, and start fraction, 3, divided by, 4, end fraction of 8 is 6.
  • The vertical displacement is 4 units, and start fraction, 3, divided by, 4, end fraction of 4 is 3.

How do we divide a line segment into two segments with a certain ratio of lengths given the coordinates of the endpoints?

Let's start by defining our variables. We'll call the coordinates of the endpoints A and B. We'll also call the ratio of the lengths r.
Now, let's think about what we're trying to do. We want to find a point P on the line segment that divides it into two segments, A, P and P, B, with lengths that have a ratio of r.
We can use a formula to find the coordinates of P:
open bracket, P, equals, start fraction, r, B, plus, A, divided by, r, plus, 1, end fraction, close bracket
This formula works in any dimension, so we can use it for both 2D and 3D line segments.
To recap, we need to know the coordinates of the two endpoints, A and B, and the ratio of the lengths, r. We plug those values into the formula, and we'll get the coordinates for P.

What's the difference between parallel and perpendicular lines?

Parallel lines are two lines that will never intersect. They have the same slope, or steepness. Perpendicular lines, on the other hand, intersect at a 90 degree angle. Their slopes are opposite reciprocals of each other. For example, if one line has the slope start fraction, 2, divided by, 3, end fraction, the perpendicular line will have the slope minus, start fraction, 3, divided by, 2, end fraction.

How do we find the equation of a parallel or perpendicular line?

To find the equation of a parallel line, we'll use the same slope as the original line, but a different y-intercept. If we know a point left parenthesis, x, start subscript, 1, end subscript, comma, y, start subscript, 1, end subscript, right parenthesis through which the parallel line passes, we can substitute the coordinates into the point-slope formula, where m represents the slope of the line.
y, minus, y, start subscript, 1, end subscript, equals, m, left parenthesis, x, minus, x, start subscript, 1, end subscript, right parenthesis
To find the equation of a perpendicular line, we'll use the opposite reciprocal slope, and then use the point-slope formula to find the y-intercept.

Where are these topics used in the real world?

Cartographers (map-makers) often use coordinate geometry to draw accurate maps. Architects and engineers use these concepts when designing buildings and bridges. Computer graphics designers use coordinate geometry to create realistic 3D images.

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