9.5 Coordinate Geometry

State the gradient $m$ and the $y$-intercept $c$ of the line $y = 3x + 2$.

Find the gradient of the line through $(1, 2)$ and $(4, 8)$.

Find the midpoint of the segment from $(2, 4)$ to $(8, 10)$.

Find the distance between $(0, 0)$ and $(3, 4)$.

A line parallel to $y = 4x + 7$. Find its gradient.

A line perpendicular to $y = 2x + 3$. Find its gradient.

Complete the table of values for $y = 2x + 1$ at $x = -1, 0, 1, 2$.

Write the equation of the line with gradient 3 and $y$-intercept -4.

State the equation of the horizontal line through $(2, 7)$ and the vertical line through $(-3, 5)$.

Find the $x$- and $y$-intercepts of $y = 3x + 6$.

Find the gradient through $(3, 8)$ and $(7, 2)$.

Find the distance between $(1, 1)$ and $(4, 5)$ as a simplified surd.

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

Find the equation of the line parallel to $y = 2x + 3$ passing through $(1, 4)$.

Find the equation of the line perpendicular to $y = 2x + 1$ through $(4, 3)$.

Find the midpoint of $A(-4, 3)$ and $B(6, -7)$ with mixed signs.

Find the equation of the line with gradient 3 through $(2, 1)$.

Decide parallel / perpendicular / neither: (a) $y = 3x + 2$ and $y = 3x - 5$. (b) $y = 4x + 1$ and $y = -x/4 + 5$.

Find the equation of the line perpendicular to $y = (1/3)x + 2$ passing through $(3, 1)$.

Find the equation of the line through (2, 4) and (5, 4).

A line passes through $A(-2, 1)$ and $B(7, 4)$. Find its equation.

The point C divides AB in ratio 2:1, $A(-2, 1)$, $B(7, 4)$. Find C.

Find the equation of the perpendicular bisector of A(1, 2) and B(5, 6).

Find the $x$- and $y$-intercepts of $3x + 4y = 12$.

Does the point (3, 7) lie on the line $y = 2x + 1$? Justify.

Find the equation of the line parallel to $2x + 3y = 6$ passing through $(0, 0)$.

Find the perimeter of triangle $A(0, 0)$, $B(4, 3)$, $C(8, 0)$.

A line $\ell_1$ has equation $y = mx + 2$. It is perpendicular to $\ell_2: y = (1/4)x + 5$. Find $m$.

Find the foot of the perpendicular from $P(5, 1)$ to the line $y = 2x - 1$.

A line through $(1, 5)$ is parallel to a line through $(0, 1)$ and $(4, 9)$. Find its equation.

A triangle has vertices A(1, 2), B(7, 4), C(4, 10). (a) Find the midpoint M of BC. (b) Find the equation of the median from A.

Show that points P(-1, -3), Q(2, 3), R(5, 9) are collinear.

Find the perpendicular distance from $P(5, 1)$ to the line $y = 2x - 1$.

A line passes through $(a, 0)$ and $(0, b)$. Find its equation.

Find the area of the triangle bounded by $y = x + 1$, $y = -x + 5$, $y = 0$.

A triangle has vertices $A(0, 0), B(4, 2), C(1, 5)$. Verify whether it is right-angled.

The point M(3, 2) is the midpoint of AB, where A = (1, 5). Find B.

A quadrilateral has vertices A(0, 0), B(4, 0), C(5, 3), D(1, 3). Find the gradients of all 4 sides and identify the shape.

Find the value of $k$ for which the lines $y = 2x + k$ and $y = 3x + 1$ intersect at a point on the $x$-axis.

The line $\ell$ has equation $3x - 4y = 12$. Find (a) the slope, (b) the $x$- and $y$-intercepts, (c) the perpendicular distance from origin to $\ell$.