Distance, section formula, and slope

The Cartesian plane assigns to each point $P$ a pair $(x, y)$ , the coordinates. Every geometric quantity (distance, midpoint, slope) becomes an algebraic computation on these coordinates. This subtopic introduces the three workhorses you will use in every later subtopic.

Definitions

• Distance between $P(x_1, y_1)$ and $Q(x_2, y_2)$: $|PQ| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.$ This is just the Pythagorean theorem applied to the right triangle with horizontal leg $|x_2 - x_1|$ and vertical leg $|y_2 - y_1|$.

• Section formula (internal division). If $R$ divides $PQ$ internally in the ratio $m : n$, then $R = \left(\frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n}\right).$

• External division. If $R$ divides $PQ$ externally in the ratio $m : n$ (so $R$ lies outside segment $PQ$), replace $n$ by $-n$: $R = \left(\frac{m x_2 - n x_1}{m - n}, \frac{m y_2 - n y_1}{m - n}\right).$

• Midpoint (special case $m = n = 1$): $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right).$

• Slope of a line through $P(x_1, y_1)$ and $Q(x_2, y_2)$ (with $x_1 \ne x_2$): $m = \frac{y_2 - y_1}{x_2 - x_1}.$

If $x_1 = x_2$, the line is vertical and slope is undefined (sometimes said to be "infinite").

Geometric meaning of slope

The slope is the tangent of the angle $\theta$ that the line makes with the positive $x$-axis (measured counter-clockwise): $m = \tan\theta$. A line going up from left to right has positive slope; going down has negative slope; horizontal has slope $0$.

Parallel lines have equal slopes. Perpendicular lines have slopes whose product is $-1$ (provided neither is vertical).

Worked examples

Example 1. Distance between $(3, 4)$ and $(-1, 2)$.

$\sqrt{(3 - (-1))^2 + (4 - 2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}$.

Example 2. Find the point dividing the segment from $(1, 2)$ to $(7, 11)$ in the ratio $2 : 1$ internally.

$R = \left(\dfrac{2 \cdot 7 + 1 \cdot 1}{3}, \dfrac{2 \cdot 11 + 1 \cdot 2}{3}\right) = \left(\dfrac{15}{3}, \dfrac{24}{3}\right) = (5, 8)$.

Example 3. Midpoint of $(-2, 5)$ and $(4, -3)$.

$M = \left(\dfrac{-2 + 4}{2}, \dfrac{5 + (-3)}{2}\right) = (1, 1)$.

Example 4. Slope of the line through $(2, 3)$ and $(5, 9)$.

$m = \dfrac{9 - 3}{5 - 2} = \dfrac{6}{3} = 2$.

Example 5. Show that $(1, 2)$, $(4, 5)$, $(7, 8)$ are collinear.

Slope $(1,2) \to (4,5)$: $\dfrac{5 - 2}{4 - 1} = 1$. Slope $(4,5) \to (7,8)$: $\dfrac{8 - 5}{7 - 4} = 1$. Same slope, common point , collinear.

Try it yourself

1. Distance between $(0, 0)$ and $(6, 8)$.
2. Distance between $(-5, 3)$ and $(7, -2)$.
3. Find $R$ dividing $A(2, -3)$ and $B(8, 9)$ in ratio $1 : 2$ internally.
4. Find $R$ dividing $A(1, 4)$ and $B(7, -2)$ externally in ratio $3 : 1$.
5. Midpoint of $(p, q)$ and $(p + 4, q - 6)$.
6. Slope of the line joining $(-3, 5)$ and $(4, -2)$.
7. If three points $A, B, C$ have slopes of $AB$ and $BC$ equal, what can you conclude?
8. Find the coordinates of the points that trisect the segment from $(1, 1)$ to $(7, 13)$.
9. Are the points $(-2, 1), (3, 3), (8, 5)$ collinear?
10. Find $x$ such that the slope of the line through $(x, 4)$ and $(1, 2)$ is $3$.
11. Show that the points $A(1, 1)$, $B(4, 5)$, $C(8, 8)$ form a right triangle.
12. Find the length of the median from $A(2, 5)$ to side $BC$ where $B = (4, 7)$ and $C = (-2, 3)$.

Pitfalls / Tricks

• For a vertical line ($x_1 = x_2$), slope is undefined , not zero.
• The section formula gives the coordinates with the same orientation as $P$ to $Q$. Be careful which point is which.
• For external division, the formula has a minus sign.
• Insight. Every geometric question , collinearity, perpendicularity, midpoint, division , has a one-line algebraic test. Knowing those tests is half the chapter.