Problem 33: Plot (3,4) and (-2,5) on the Cartesian plane.

The distance $D$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

Let $(x_1, y_1) = (0,0)$ and $(x_2, y_2) = (6,8)$.

: Find the midpoint of the line joining (2,3) and (6,7).

The midpoint $M$ of a line segment joining $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

Let $(x_1, y_1) = (2,3)$ and $(x_2, y_2) = (6,7)$.

: Find the gradient of line AB if A is (2,3) and B is (4,1).

The gradient (slope) $m$ of a line passing through $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

Let $(x_1, y_1) = (2,3)$ and $(x_2, y_2) = (4,1)$.

: Find the equation of the Line if gradient/slope = 2 and it passes through (2,5).

The point-slope form is $y - y_1 = m(x - x_1)$, where $m$ is the gradient and $(x_1, y_1)$ is a point on the line.

Given $m = 2$ and $(x_1, y_1) = (2,5)$.

: The vertices of a triangle are (1,1), (4,1) and (2,5). Find its area.

Notice that two vertices, $(1,1)$ and $(4,1)$, have the same y-coordinate. This means the side connecting these two points is a horizontal base.
The length of the base ($b$) is the absolute difference of their x-coordinates:

The height ($h$) of the triangle is the perpendicular distance from the third vertex $(2,5)$ to this base. This is the absolute difference between the y-coordinate of the third vertex and the y-coordinate of the base: