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Analytic geometry FAQ

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 $\left(0,0\right)$, point $C$ is at $\left(8,4\right)$, and we want to find point $B$ such that $AB$ is $\frac{3}{4}$ of $AC$, we can use a strategy like this:
• The horizontal displacement is $8$ units, and $\frac{3}{4}$ of $8$ is $6$.
• The vertical displacement is $4$ units, and $\frac{3}{4}$ 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, $AP$ and $PB$, with lengths that have a ratio of $r$.
We can use a formula to find the coordinates of $P$:
$\left[P=\frac{rB+A}{r+1}\right]$
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 $\frac{2}{3}$, the perpendicular line will have the slope $-\frac{3}{2}$.

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 $\left({x}_{1},{y}_{1}\right)$ through which the parallel line passes, we can substitute the coordinates into the point-slope formula, where $m$ represents the slope of the line.
$y-{y}_{1}=m\left(x-{x}_{1}\right)$
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.

Want to join the conversation?

• Where is the formula derived that gives the location of P? I have not seen it before, and have done all the analytic geometry content. Does this formula also extend to alternative coordinate systems, like polar(would you just multiply the angle by the ratio?)? Thanks in advance!
• Can anyone give an example on "the formula that locates the coordinates of P"?, thank you:)
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• andrew tate
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