1. **Problem Statement:** We are given two points on a Cartesian coordinate system: $ (x_1, y_1) $ and $ (x_2, y_2) $. We want to find: - The distance between these two points. - The gradient (slope) of the line segment connecting them. - The coordinates of the midpoint of the line segment. - Application: Calculate the area of a geometric figure using these concepts. 2. **Distance Formula:** The distance $d$ between two points $ (x_1, y_1) $ and $ (x_2, y_2) $ is given by the formula derived from the Pythagorean theorem: $$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$ This formula calculates the length of the line segment connecting the two points. 3. **Gradient (Slope) Formula:** The gradient $m$ of the line segment connecting the points is the ratio of the vertical change to the horizontal change: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ Important rules: - If two lines are parallel, their gradients are equal. - If two lines are perpendicular, the product of their gradients is $-1$, i.e., $m_1 \times m_2 = -1$. 4. **Midpoint Formula:** The midpoint $M$ of the line segment joining the points is the average of the $x$-coordinates and the $y$-coordinates: $$ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$ This point lies exactly halfway between the two points. 5. **Area Application:** Using coordinates, the area of a triangle formed by points $ (x_1, y_1) $, $ (x_2, y_2) $, and $ (x_3, y_3) $ can be calculated by: $$ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| $$ This formula uses the coordinates to find the absolute value of the determinant, giving the area. **Summary:** - Distance measures length between points. - Gradient measures slope and helps identify parallel/perpendicular lines. - Midpoint finds the center point between two points. - Area formula applies coordinate geometry to find areas of polygons. These formulas are fundamental in analytical geometry and help represent and analyze geometric figures on the Cartesian plane.