Exercises: Equation of a line and area of a triangle

Find the equation of the line through the point $(2, 3)$ with slope $4$.

Find the equation of the line through the points $A(1, 2)$ and $B(5, 10)$.

Find the points where the line $3x + 2y = 12$ cuts the $x$-axis and the $y$-axis.

A taxi in Cork charges a fixed fee plus a rate per kilometre. A 5 km journey costs €11 and a 12 km journey costs €25.

1. a) Find the equation of the line relating cost $C$ to distance $d$.
2. b) State the fixed fee and the rate per kilometre.
3. c) Find the cost of a 20 km journey.

The triangle $ABC$ has vertices $A(0, 0)$, $B(6, 0)$ and $C(2, 5)$.

1. a) Calculate the area of the triangle using the formula $\frac{1}{2}|x_1y_2 - x_2y_1|$ with a suitable translation, or otherwise.
2. b) Find the equation of the line $BC$.

The line $l$ has equation $2x - 5y + 10 = 0$.

1. a) Find the slope of $l$ and the points where it cuts the axes.
2. b) Find the equation of the line through $(3, -1)$ that is parallel to $l$.
3. c) Find the equation of the line through $(3, -1)$ that is perpendicular to $l$.

The points $A(-1, 2)$, $B(5, 4)$ and $C(3, -4)$ are the vertices of a triangle.

1. a) Find the equation of the line $AB$.
2. b) Find the area of triangle $ABC$.
3. c) Find the perpendicular distance from $C$ to $AB$, correct to two decimal places.

The line $l$ passes through $(1, 6)$ and cuts the positive $x$-axis and positive $y$-axis at $P$ and $Q$ respectively. The triangle $OPQ$, where $O$ is the origin, has area $12$ square units and $P$ is at $(p, 0)$.

1. a) Write down the coordinates of $Q$ in terms of $p$ and the slope condition from the point $(1, 6)$.
2. b) Show that $p$ satisfies a quadratic equation and find its possible values.
3. c) Write down the equation of $l$ for each value of $p$ found.

Three points $A(1, 1)$, $B(4, 7)$ and $C(k, 13)$ are given.

1. a) Find the value of $k$ for which $A$, $B$ and $C$ are collinear, and justify using slopes.
2. b) Using this value, explain why the area of triangle $ABC$ is zero.
3. c) Find the two values of $k$ for which triangle $ABC$ has area $15$ square units.