Distance and Section Formulas: Quick Revision

A brief note on Distance and Section Formulas. Read it before a test. Try each question at the end before you open its answer.

The formulas

• Distance: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
• Midpoint: $\left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)$
• Ratio $m : n$: $\left(\dfrac{m x_2 + n x_1}{m + n}, \dfrac{m y_2 + n y_1}{m + n}\right)$
• Centroid: the average of the three corners
• Area of a triangle: $\tfrac{1}{2}\left|S\right|$, where $S = x_1(y_2 - y_3)$ $+ x_2(y_3 - y_1)$ $+ x_3(y_1 - y_2)$

Using them

• What kind of shape? Compare the squares of the sides, and of the diagonals for a quadrilateral.
• Collinear points make a triangle of area $0$.
• Points of trisection use the ratios $1 : 2$ and $2 : 1$.

A worked example

$A(-1, 0)$ and $B(7, 8)$. Find $AB$, the midpoint, and the point that divides $AB$ in the ratio $1 : 3$.

1. Length. $\sqrt{8^2 + 8^2} = \sqrt{128} = 8\sqrt{2} \approx 11.31$ units.
2. Midpoint. $\left(\dfrac{-1 + 7}{2}, \dfrac{0 + 8}{2}\right) = (3, 4)$.
3. Ratio $1 : 3$. $\left(\dfrac{7 + 3 \times (-1)}{4}, \dfrac{8 + 3 \times 0}{4}\right) = (1, 2)$.

Practice with new numbers

A program to try

The program works out the worked example.

Mistakes to avoid

• Adding instead of subtracting inside the distance formula.
• Crossing the weights in the section formula: $m$ goes with the far end, $B$.
• Halving the difference instead of the sum for a midpoint.
• Forgetting the half in the area of a triangle.

Check yourself

1. What is the distance between $(-1, 3)$ and $(2, 7)$?

2. What is the midpoint of $(4, -6)$ and $(-2, 0)$?

3. What is the centroid of $(0, 0)$, $(9, 0)$ and $(0, 6)$?

4. What is the area of the triangle $(0, 0)$, $(5, 0)$, $(5, 4)$?