Distance between two points
A radar, a GPS or a video game constantly needs to know "how far is this from that?". On the Cartesian plane that question is always answered in the same way: with Pythagoras' theorem.
The distance formula
Calculate the exact distance between the points $A(2, 3)$ and $B(5, 7)$ in the Cartesian plane.
Calculate the exact distance between the points $A(2, 3)$ and $B(5, 7)$ in the Cartesian plane.
A ship is at coordinates $P(-4, 1)$ and a lighthouse at $Q(2, -7)$. If each unit on the plane equals one kilometre, what is the straight-line distance from the ship to…
A ship is at coordinates $P(-4, 1)$ and a lighthouse at $Q(2, -7)$. If each unit on the plane equals one kilometre, what is the straight-line distance from the ship to the lighthouse?
Find the length of the segment whose endpoints are the origin of coordinates $O(0, 0)$ and the point $M(-8, 6)$.
Find the length of the segment whose endpoints are the origin of coordinates $O(0, 0)$ and the point $M(-8, 6)$.
Perimeters and classifying triangles
With the distance formula you can calculate the perimeter of any polygon (by adding up the lengths of its sides) and check whether a triangle is equilateral, isosceles or right-angled, without needing to draw it to scale.
The vertices of a triangle are $A(1, 1)$, $B(4, 1)$ and $C(1, 5)$. Calculate the perimeter of this triangle by adding the lengths of its three sides.
The vertices of a triangle are $A(1, 1)$, $B(4, 1)$ and $C(1, 5)$. Calculate the perimeter of this triangle by adding the lengths of its three sides.
On an architectural plan, three main columns will be situated at coordinates $P(-2, 1)$, $Q(2, 4)$ and $R(5, 0)$. Prove mathematically whether the triangle formed by…
On an architectural plan, three main columns will be situated at coordinates $P(-2, 1)$, $Q(2, 4)$ and $R(5, 0)$. Prove mathematically whether the triangle formed by these three columns is isosceles.
Prove, by calculating the lengths of the three sides and using Pythagoras' theorem, that the triangle with vertices $A(2, 1)$, $B(6, 1)$ and $C(2, 4)$ is a right-angled…
Prove, by calculating the lengths of the three sides and using Pythagoras' theorem, that the triangle with vertices $A(2, 1)$, $B(6, 1)$ and $C(2, 4)$ is a right-angled triangle.
Finding points at a given distance
Sometimes the distance is the given information and the unknown is a point: in that case you set up the distance equation and solve it.
The distance between points $A(1, 2)$ and $B(x, 5)$ is exactly $5$ units. Find the two possible values of the coordinate $x$.
The distance between points $A(1, 2)$ and $B(x, 5)$ is exactly $5$ units. Find the two possible values of the coordinate $x$.
Find the coordinates of all points on the x-axis (axis of abscissas) that are at a distance of exactly $10$ units from the point $M(-6, 8)$.
Find the coordinates of all points on the x-axis (axis of abscissas) that are at a distance of exactly $10$ units from the point $M(-6, 8)$.
A straight pipe must connect a well located at point $A(-3, 2)$ to a water pump $P$ that must be installed on the y-axis (axis of ordinates). If pressure constraints…
A straight pipe must connect a well located at point $A(-3, 2)$ to a water pump $P$ that must be installed on the y-axis (axis of ordinates). If pressure constraints require the length of the pipe to be exactly $\sqrt{29}$ units, find the possible coordinates for the pump $P$.
Collinearity and equidistant points
The vertices of a triangular plot of land are $A(-4, -2)$, $B(2, 6)$ and $C(6, 3)$. First prove that the land forms a right-angled triangle, and then calculate its…
The vertices of a triangular plot of land are $A(-4, -2)$, $B(2, 6)$ and $C(6, 3)$. First prove that the land forms a right-angled triangle, and then calculate its perimeter and its exact area.
An automated irrigation system consists of a circular sprinkler centred at $C(x, y)$ that must equally water three trees located at coordinates $T_1(0, 0)$, $T_2(6, 0)$…
An automated irrigation system consists of a circular sprinkler centred at $C(x, y)$ that must equally water three trees located at coordinates $T_1(0, 0)$, $T_2(6, 0)$ and $T_3(0, 8)$. Determine the coordinates where the sprinkler $C$ should be placed and what its exact coverage radius will be.
Determine whether the points $A(-2, 1)$, $B(1, 3)$ and $C(7, 7)$ are collinear. To do this, calculate the distances $d(A,B)$, $d(B,C)$ and $d(A,C)$, and check whether…
Determine whether the points $A(-2, 1)$, $B(1, 3)$ and $C(7, 7)$ are collinear. To do this, calculate the distances $d(A,B)$, $d(B,C)$ and $d(A,C)$, and check whether the equality $d(A,B) + d(B,C) = d(A,C)$ holds.
Back to the beginning: If a radar at the origin $(0,0)$ detects a ship at $(12, 5)$ km, the direct distance in a straight line is found with the formula: $d = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13$ km.