Prove that if two lines with slopes m1 and m2 are...

Prove that if two lines with slopes $m_1$ and $m_2$ are perpendicular, then $m_1 m_2=-1$. (Fig. 10-6.) The slopes of the lines must have opposite signs. In the figure, $m_1$ is chosen (arbitrarily) positive and $m_2$ is chosen negative. (FIGURE CAN'T COPY) Since $l_1$ has slope $m_1$, a run of 1 (segment $P B$ ) yields a rise of $m_1$ along $l_1$ (segment $C B$ ). Similarly, since $l_2$ has slope $m_2$, a run of 1 yields a (negative) rise of $m_2$ along $l_2$, thus segment $A B$ has length $-m_2$. Since the lines are perpendicular, the triangles $P C B$ and $A P B$ are similar. Hence ratios of corresponding sides are equal; it follows that $$ \begin{gathered} \frac{C B}{P B}=\frac{P B}{A B} \\ \frac{m_1}{1}=\frac{1}{-m_2} \\ m_1 m_2=-1 \end{gathered} $$

Prove that if two lines with slopes $m_1$ and $m_2$ are perpendicular, then $m_1 m_2=-1$. (Fig. 10-6.)
The slopes of the lines must have opposite signs. In the figure, $m_1$ is chosen (arbitrarily) positive and $m_2$ is chosen negative.
(FIGURE CAN'T COPY)
Since $l_1$ has slope $m_1$, a run of 1 (segment $P B$ ) yields a rise of $m_1$ along $l_1$ (segment $C B$ ). Similarly, since $l_2$ has slope $m_2$, a run of 1 yields a (negative) rise of $m_2$ along $l_2$, thus segment $A B$ has length $-m_2$. Since the lines are perpendicular, the triangles $P C B$ and $A P B$ are similar. Hence ratios of corresponding sides are equal; it follows that
$$
\begin{gathered}
\frac{C B}{P B}=\frac{P B}{A B} \\
\frac{m_1}{1}=\frac{1}{-m_2} \\
m_1 m_2=-1
\end{gathered}
$$

Key Concepts

Slope

Slope is a measure of a line's steepness, defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. This concept is fundamental in coordinate geometry because it quantifies the incline or decline of a line.

Perpendicular Lines

In the coordinate plane, two lines are perpendicular if they intersect to form a right angle (90 degrees). A significant property of perpendicular lines is that the product of their slopes is -1, which arises from the relationship between their directional changes.

Similar Triangles

Similar triangles have corresponding angles that are equal and corresponding sides in proportional lengths. This property is used to relate different segments created by intersecting lines, establishing proportional relationships that are key to deriving algebraic conclusions such as those involving slopes.

Ratios and Proportions in Geometry

Ratios are comparisons between two quantities, and proportions are equations that state two ratios are equal. In the context of coordinate geometry, proportions derived from similar triangles enable one to set up equations that relate slopes and distances, which is critical in proofs involving relationships like the one for perpendicular lines.

Transcript

00:01 The first thing you want to do when considering this problem is don't even pay attention to the x and y axis.

00:07 We don't need that.

00:08 Don't use it.

00:10 And just assume that p is our point of origin, or this is where 0 -0 is located.

00:17 And so this will help us because then it becomes really obvious that q is located at the point 0 -1, because we change the origin for p to be 0 -0.

00:31 Then the equation of the line pm, which is right here, this line pm, it's going to be y equals m1x, because that is the slope of pm is m1 over 1, because if that helps you can do it over 1, x.

00:58 Then we see that m is going to equal m1x, comma x, which equals m1x.

01:07 Which implies that mq is the same thing, the length of mq.

01:15 You're wondering what mq is, is this line right here? it's just m1.

01:21 You can make a similar argument for b, which is asking pretty much the same thing.

01:27 Confirm that m2 is the length of qn.

01:34 But notice that when you do this argument, m2 is below the 0 -0.

01:39 So let me do really light tracing the new x and y axis, if it helps you a little bit.

01:47 Here is your new x and y axis.

01:56 And notice that m2 is below so that the length or the distance of m2 is going to be negative m2.

02:06 So just be careful when you take a look at that.

02:09 All right, i'm going to erase this so i have some space to work for the next piece, which is part c...

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