Coordinate Geometry

Which of the following points is closest to the origin?

A rectangle has vertices $O(0,0)$, $A(5,0)$, $B(5,3)$, and $C(0,3)$. What is the perimeter of this rectangle?

What is the abscissa of any point lying on the $y$-axis?

Find the coordinates of the midpoint of the line segment joining $O(0, 0)$ and $R(6, 8)$.

In the given coordinate plane, a square $ABCD$ is represented such that its diagonal $AC$ is part of the line $y = x$. If the coordinates of point $A$ are $(1, 1)$ and the coordinates of point $C$ are $(5, 5)$, find the coordinates of the other two vertices $B$ and $D$ of the square.

A point $M$ moves such that its distance from the $y$-axis is always 3 units. What is the equation representing the path of point $M$ in the diagram?

In the following diagram, if $ABCD$ is a rhombus with vertices $A(3, 0)$, $B(4, 5)$, $C(-1, 4)$, and $D(-2, -1)$, what is the length of its diagonal $AC$?

A line segment $PQ$ has length 10 units. If $P$ is at $(2, -3)$ and the abscissa of $Q$ is $10$, find the possible values of the ordinate of $Q$.

The points $A(2, -2)$, $B(-2, 1)$, and $C(5, 2)$ are the vertices of a triangle. Determine whether the triangle is isosceles, scalene, or right-angled.

A circle is drawn with its center at the origin $O(0, 0)$. If a point $A(4, 3)$ lies on the circle, which of the following points also lies on the same circle?

The distance between $(0, a)$ and $(0, b)$ is:

Find the distance between the points $(k, 0)$ and $(0, k)$.

Which of the following points is at a distance of $13$ units from the origin?

Find the distance of the point $P(3, 4)$ from the y-axis.

If the point $A(x, 2)$ is equidistant from $B(8, -2)$ and $C(2, -2)$, find the value of $x$.

Determine the perimeter of a triangle with vertices $O(0, 0)$, $P(5, 0)$, and $Q(0, 12)$.

Find the relation between $x$ and $y$ such that the point $P(x, y)$ is equidistant from the points $A(7, 1)$ and $B(3, 5)$.

If the distance between the points $(x, -1)$ and $(3, 2)$ is $5$ units, find the product of all possible values of $x$.

The vertices of a triangle are $A(3, 0)$, $B(6, 4)$, and $C(-1, 3)$. Calculate the lengths of the sides and determine if the triangle is isosceles.

A circle has its center at $C(2a, a-7)$. Find the value of $a$ if the circle passes through the point $P(11, -9)$ and has a radius of $5\sqrt{2}$ units.

A line segment $AB$ joins the points $A(1, -3)$ and $B(4, 6)$. A point $P$ lies on this segment such that it divides $AB$ internally in the ratio $2:1$. Find the coordinates of point $P$.

Find the ratio in which the y-axis divides the line segment joining $A(-4, 3)$ and $B(2, 5)$.

If the point $P(k, 0)$ divides the segment joining $A(2, -2)$ and $B(-7, 4)$ in the ratio $1:2$ internally, find the value of $k$.

The midpoint of the line segment joining $(2a, 4)$ and $(-2, 2b)$ is $(1, 2a+1)$. Find the value of $b$.

The three vertices of a rhombus taken in order are $(3, 0)$, $(4, 5)$, $(-1, 4)$. Find the coordinates of the fourth vertex.

If $A(-2, -2)$ and $B(2, -4)$ are two points, find the coordinates of $P$ such that $AP = \frac{3}{7} AB$ and $P$ lies on the line segment $AB$.

In what ratio does the point $P(-4, 6)$ divide the line segment joining the points $A(-6, 10)$ and $B(3, -8)$?

A surveyor identifies two markers at $L(4, 5)$ and $K(-2, -7)$. A new boundary point $B$ is set on the line segment $LK$ such that $LB = \frac{2}{3} LK$. Find the coordinates of $B$.

A marathon track is mapped on a coordinate plane. The hydration stations are at $A(-8, -2)$ and $B(2, 8)$. If a medical tent $M$ is placed such that it divides $AB$ in the ratio $1:4$ internally, find the coordinates of $M$.

Find the coordinates of the point $R$ which is three-quarters of the way from $P(1, 7)$ to $Q(5, -1)$.

If a point's y-coordinate is $4$ and it lies on the y-axis, what is its x-coordinate?

Which of the following points lies in Quadrant II?

The abscissa of all points on the y-axis is:

What is the distance of the point $(0, -7)$ from the x-axis?

If the distance between $P(x, -1)$ and $Q(3, 2)$ is $5$, then one of the possible values of $x$ is:

The area of a triangle with vertices $(a, 0)$, $(0, b)$, and $(0, 0)$ is:

The coordinates of a point $P$ which divides the line segment joining $A(1, 3)$ and $B(4, 6)$ in the ratio $2:1$ internally are:

Three vertices of a triangle are $A(x, y)$, $B(1, 2)$, and $C(2, 1)$. If the centroid of the triangle is at the origin $(0, 0)$, what are the coordinates of vertex $A$?

A hiker starts at point $A(2, 3)$ and reaches $B(10, 9)$. The hiker stops for a break at point $M$, which is the midpoint of $AB$. From $M$, the hiker travels to a point $C$ such that $MC$ is perpendicular to $AB$ and the distance $MC = 5$ units. If $C$ is in the first quadrant and 'above' $AB$, what are the coordinates of $C$?

In a city grid, a square park has two opposite vertices at $A(-1, 2)$ and $C(3, 2)$. What is the length of the side of this square park?

Calculate the distance between $L(0, 5)$ and $M(0, -3)$.

In the square $ABCD$, if $A$ is $(1, 1)$ and $B$ is $(4, 1)$, what is the length of side $AB$?

Determine the distance between $P(-2, -2)$ and $Q(2, 2)$.

Find the distance between the points $A(a, 0)$ and $B(0, a)$ where $a > 0$.

Given a triangle with vertices $A(-2, 0)$, $B(2, 0)$ and $C(0, 2\sqrt{3})$, what type of triangle is it based on the distances between its vertices?

What is the radius of a circle centered at $(2, 3)$ that passes through the point $(5, 7)$?

A point $P$ is at a distance of $\sqrt{10}$ units from the origin. If the y-coordinate of $P$ is twice its x-coordinate and $P$ is in the first quadrant, find $P$.

Two athletes are running on a coordinate field. Athlete 1 is at $A(4, 3)$ and Athlete 2 is at $B(x, 1)$. If the distance between them is $2\sqrt{5}$ units and $x < 4$, find the coordinates of Athlete 2.

A circle has its center at $C(2a, a-7)$. If the circle passes through the point $P(11, -9)$ and has a radius of $5\sqrt{2}$ units, find the value of $a$ if $a > 0$.

If the point $P(x, y)$ is equidistant from $A(5, 1)$ and $B(-1, 5)$, which of the following linear relationships correctly describes the locus of $P$?

Find the coordinates of the midpoint of the line segment joining the points $A(2, -4)$ and $B(6, 2)$ as shown in the coordinate plane.

If point $M$ is the midpoint of the line segment joining $C(-2, 8)$ and $D(-6, -4)$, find the coordinates of $M$.

Find the point on the x-axis which is equidistant from $(2, -5)$ and $(-2, 9)$.

What are the coordinates of the point $P$ that divides the line segment $A(x_1, y_1)$ and $B(x_2, y_2)$ externally in the ratio $m:n$?

The coordinates of the fourth vertex $D$ of a parallelogram $ABCD$ whose three vertices are $A(-2, 3)$, $B(6, 7)$ and $C(8, 3)$ are:

If the midpoint of the line segment joining $A(3, 4)$ and $B(k, 6)$ is $(x, y)$ and $x + y - 10 = 0$, find $k$.

The ratio in which the line $x - y - 2 = 0$ divides the line segment joining $(3, -1)$ and $(8, 9)$ is:

If $A(2, 3)$, $B(4, k)$, and $C(6, -3)$ are the vertices of a triangle whose area is $0$ (points are collinear), find the value of $k$. What is the ratio in which $B$ divides $AC$?

In what ratio does the line $x - y - 2 = 0$ divide the line segment joining the points $A(3, -1)$ and $B(8, 9)$?

Find the ratio in which the point $P(m, 6)$ divides the join of $A(-4, 3)$ and $B(2, 8)$. Also, find the value of $m$.