Coordinate Geometry

A line has a rise of $10$ units for a run of $5$ units. What is its gradient?

Calculate the gradient of the line passing through $(2, 3)$ and $(4, 7)$.

What is the gradient of the line represented by $y = -x + 7$?

A line segment connects $A(0, 5)$ and $B(5, 0)$. What is its gradient?

Three points $A(1, 2)$, $B(3, 6)$, and $C(5, y)$ are collinear. Find the value of $y$.

What is the gradient of the line that passes through the points $(a, b)$ and $(2a, 3b)$?

A line $L_1$ has a gradient of $2/3$. Line $L_2$ is perpendicular to $L_1$. What is the gradient of $L_2$?

Points $A(-2, -5)$, $B(2, 3)$, and $C(4, 7)$ lie on a coordinate plane. Determine the gradient of the line passing through these points and state if they are collinear.

A square has vertices $(0, 0)$, $(a, 0)$, $(a, a)$, and $(0, a)$. A line is drawn from the origin to the midpoint of the side connecting $(a, a)$ and $(0, a)$. What is the gradient of this line?

The gradient of the line segment joining $(k+1, 4)$ and $(3, k-2)$ is $1$. Find the possible value(s) of $k$.

What is the gradient of the horizontal line shown in the diagram?

A line passes through the origin $(0,0)$ and the point $(4, 2)$ as shown. What is the equation of this line?

Find the $y$-intercept of the line shown in the diagram.

The diagram shows a straight line $L$. What is the gradient of line $L$?

The diagram shows a line passing through point $P(2, 5)$. If the gradient of the line is $-1$, find the coordinates where the line intercepts the $y$-axis.

The line in the diagram has the equation $2y = 4x + 6$. What is the gradient $m$ and the $y$-intercept $c$ when written in the form $y = mx + c$?

A line crosses the $x$-axis at $(4, 0)$ and the $y$-axis at $(0, 3)$. What is the equation of this line?

A surveyor identifies two points on a map, $A(-2, -5)$ and $B(6, 7)$. A path needs to be constructed such that it is the perpendicular bisector of the line segment $AB$. Determine the equation of the line representing this path.

A civil engineer is designing a straight access road that must pass through a supply depot located at point $P(4, 11)$. The road must also be perpendicular to an existing pipeline represented by the equation $2y + 8x = 15$. Find the equation of the line representing the new road in the form $y = mx + c$.

The cost of renting a car, $C$, is a linear function of the distance driven, $d$, in kilometers. The base fee is represented by the $y$-intercept. If a 100 km trip costs $80 and a 250 km trip costs$155, find the equation relating $C$ and $d$, and determine the cost of a 400 km trip.

Given the endpoints $E(-4, 0)$ and $F(4, 0)$, what are the coordinates of the midpoint?

Find the distance between the points $T(-1, 1)$ and $U(2, 5)$.

Point $M$ is the midpoint of $JK$. If $J = (2, 8)$ and $K = (2, -2)$, find the coordinates of $M$.

What is the length of the segment $AB$ with coordinates $A(0, 0)$ and $B(12, 5)$?

A circle has a diameter with endpoints $D(-4, 2)$ and $E(2, 10)$. Find the coordinates of the center of the circle.

Triangle $ABC$ has vertices $A(0, 0)$, $B(6, 0)$, and $C(3, 4)$. Calculate the perimeter of the triangle.

The point $M$ is the midpoint of the line segment $XY$. If $X$ is $(2, 7)$ and $M$ is $(5, 3)$, find the coordinates of point $Y$.

Find the distance between the midpoint of $AB$ and the midpoint of $CD$, where $A(0, 0)$, $B(4, 0)$, $C(0, 6)$, and $D(0, 10)$.

The coordinates of the vertices of a parallelogram are $A(1, 2)$, $B(5, 2)$, $C(6, 5)$, and $D(2, 5)$. Find the coordinates of the point where the diagonals intersect.

A square has a diagonal with endpoints $(1, 1)$ and $(5, 5)$. What is the length of the side of the square?

The diagram shows a line $L_1$ with the equation $y = 2x + 1$. A second line, $L_2$, is parallel to $L_1$ and passes through the point $(0, -3)$. What is the equation of line $L_2$?

A line segment $AB$ is drawn on the coordinate plane. What is the gradient of a line that is perpendicular to the line passing through points $A(0, 2)$ and $B(4, 0)$?

A line $y = 4$ is parallel to which of the following lines?

The gradient of line $L_1$ is $0$. What is the gradient of a line $L_2$ that is perpendicular to $L_1$?

The line $PQ$ has the equation $y = \frac{1}{3}x + 4$. A second line, $RS$, is perpendicular to $PQ$ and passes through the point $(2, 1)$, as shown in the coordinate plane. What is the equation of the line $RS$?

In the diagram, the line $L_2$ is perpendicular to $L_1$. $L_1$ passes through $(0, 3)$ and $(3, 0)$. If $L_2$ passes through the point $(1, 2)$, what is its y-intercept?

The lines $y = ax + 7$ and $2x + 5y = 10$ are perpendicular. Calculate the value of the constant $a$.

A city planning department is designing two new roads, Road L and Road K. Road L passes through the points $A(-2, 5)$ and $B(4, 1)$. Road K is designed to be perpendicular to Road L and must pass through the point $P(2, 6)$. Find the equation of Road K in the form $ax + by = c$, where $a$, $b$, and $c$ are integers and $a > 0$.

In a navigation system, a boat is moving along the path $x - 3y + 9 = 0$. A lighthouse is located at point $H(5, 8)$. The path of a rescue drone starting from the lighthouse is perpendicular to the boat's path. Determine the coordinates of the point where the drone's path would intersect the boat's path.

A line $L$ has the equation $5x + 2y = 10$. A second line $M$ is parallel to $L$ and passes through the midpoint of the segment joining $(0, 8)$ and $(4, 0)$. Find the equation of line $M$ in the form $ax + by = c$, where $a$, $b$, and $c$ are integers.

For the function $y = x^2 + 1$, what is the value of $y$ when $x = -3$?

The graph of $y = \frac{-2}{x}$ is shown. In which quadrants does the graph lie?

The graph shows $y = (x-3)^2$. What is the coordinate of the minimum point?

Which of the following describes the shape of the graph $y = x^3 - x$ shown?

Identify the coordinates of the points where $y = x^3$ and $y = x$ intersect.

The graph of $y = a/x^2$ is shown. If the graph passes through $(1, 4)$, find the value of $a$.

Estimate the gradient of the curve $y = x^2$ at the point where $x = 1$.

The intensity of light $I$ at a distance $d$ from a source is given by $I = \frac{100}{d^2}$. If the intensity is $4$ units, what is the distance $d$?

An experimental rocket's altitude $h$ follows $h = -2t^2 + 12t$. At what two times is the rocket at an altitude of $10$ meters?

A sculptor creates a pedestal whose side profile follows $y = x^3 - 4x + 5$. At which point does the curve cross the $y$-axis?

1. Coordinates: practice question 1 for Coordinate Geometry?

1. Coordinates: practice question 2 for Coordinate Geometry?

1. Coordinates: practice question 3 for Coordinate Geometry?

1. Coordinates: practice question 4 for Coordinate Geometry?

1. Coordinates: practice question 5 for Coordinate Geometry?

1. Coordinates: practice question 6 for Coordinate Geometry?

1. Coordinates: practice question 7 for Coordinate Geometry?

1. Coordinates: practice question 8 for Coordinate Geometry?

1. Coordinates: practice question 9 for Coordinate Geometry?

1. Coordinates: practice question 10 for Coordinate Geometry?