Co-ordinate Geometry — practice questions

Find the exact distance between $P(-2, 5)$ and $Q(4, -3)$.

The points $A(0, 4)$, $B(6, 2)$ and $C(4, -4)$ form a triangle. Show that it is isosceles, giving exact side lengths, and state which vertex the equal sides meet at.

Find all points on the $y$-axis that are exactly 13 units from the point $A(5, 3)$, and explain why there are exactly two such points.

Find the midpoint of the segment joining $A(-4, 7)$ and $B(10, -3)$.

$M(4, -1)$ is the midpoint of $AB$, where $A$ is $(-3, 5)$. Find the coordinates of $B$, and verify your answer.

A triangle has vertices $A(1, 6)$, $B(7, 2)$ and $C(-1, -2)$. Show that the three medians of the triangle all pass through the point $G(\frac{7}{3}, 2)$, and state what you notice about the coordinates of $G$.

Find the gradient of the line through $(-1, 7)$ and $(3, -1)$.

Show that the points $P(-3, -5)$, $Q(0, 1)$ and $R(5, 11)$ are collinear, and explain why your method proves it.

The points $A(2, k)$, $B(5, 8)$ and $C(k, 20)$ are collinear. Find all possible values of $k$.

Find the equation of the line with gradient 3 passing through the point $(2, -1)$, in the form $y = mx + c$.

A line passes through $(-4, 1)$ and $(2, -8)$. Find its equation in the form $ax + by + c = 0$ with integer coefficients, and state its $x$- and $y$-intercepts.

The line $\ell$ has equation $3x - 4y + 12 = 0$. Find the area of the triangle formed by $\ell$ and the two coordinate axes, and determine the equation of the line through the origin that bisects this area.

Find the equation of the line through $(1, 5)$ that is perpendicular to $y = 4x - 3$.

The line $\ell$ has equation $2x + 5y = 20$. Find the equation of the line through $(4, -1)$ that is parallel to $\ell$, and the equation of the line through the same point perpendicular to $\ell$.

The points $A(0, 0)$, $B(6, 2)$, $C(8, 8)$ and $D(2, 6)$ form a quadrilateral. Determine as precisely as possible what type of quadrilateral $ABCD$ is, justifying every claim.

Find the point of intersection of the lines $y = 3x - 4$ and $y = -2x + 11$.

Two walking tracks in a reserve follow the lines $4x - 3y = 6$ and $2x + y = 13$, where units are 100 m east and 100 m north of the car park. Find where the tracks cross, and give the location in metres.

The line $y = mx + 4$ passes through the point of intersection of $2x + y = 10$ and $x - y = 2$ for one particular value of $m$. Find $m$, and determine for which values of $m$ the line $y = mx + 4$ passes through the interior of the triangle formed by those two lines and the $x$-axis.