Distance between two points

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 the lighthouse?

Find the length of the segment whose endpoints are the origin of coordinates $O(0, 0)$ and the point $M(-8, 6)$.

Cian wants to calculate the distance between the points $A(-1, 5)$ and $B(3, 2)$ in a Cartesian plane. In his notebook, he writes the following steps:

1. a) Precisely point out the conceptual and algebraic error that Cian makes in the last step and explain why the square root of a sum is not equal to the sum of the square roots.
2. b) Correct the procedure and calculate the exact numerical value of the Euclidean distance between $A$ and $B$.

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 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 triangle.

A bicycle courier leaves the shop located at $O(0,0)$, travels in a straight line to the first delivery at $A(6, 8)$ and then changes direction towards the second delivery at $B(15, -4)$. If each unit on the plane represents $100$ metres, what is the total distance travelled in kilometres?

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 $M(-6, 8)$.

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 $P$.

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 $T_1(0, 0)$, $T_2(6, 0)$ 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 the equality $d(A,B) + d(B,C) = d(A,C)$ holds.

To check whether the points $A(-1, -2)$, $B(1, 2)$, and $C(3, 6)$ are collinear on the same line, two different mathematical strategies are proposed:

• Method 1 (Metric): Calculate the three distances $d(A,B)$, $d(B,C)$, and $d(A,C)$ using the magnitude of the displacements and verify whether the sum of the two smaller ones equals the largest one.

• Method 2 (Vector): Calculate the components of vectors $\vec{AB}$ and $\vec{BC}$ and verify whether they are directly proportional.

1. a) Apply Method 1 simplifying the radicals as much as possible to work with exact values.
2. b) Apply Method 2.
3. c) Compare both approaches and argue which one is preferable from the point of view of computational economy and safety in avoiding rounding errors when distances turn out to be irrational numbers.

The base station of a cable car is at $B(0, 0)$ and the mountain peak is at $C(120, 50)$ on a two-dimensional map. A support tower needs to be installed exactly at the midpoint of the linear path. Calculate the coordinates of this midpoint and the distance from the base to the tower.

Find the coordinates of a point $P$ located on the angle bisector of the first quadrant (the line $y = x$) that is equidistant from the fixed points $A(2, 0)$ and $B(0, 4)$.

The points $A(-1, 3)$, $B(4, 5)$ and $C(2, -2)$ are three consecutive vertices of a parallelogram $ABCD$. Find the coordinates of the fourth vertex $D$ and calculate the exact length of both diagonals of the parallelogram.

Design the coordinates of three distinct points in the Cartesian plane, $A$, $B$, and $C$, all located in the first quadrant and with non-zero integer coordinates, that simultaneously satisfy the following conditions:

1. The distance between $A$ and $B$ must be exactly $\sqrt{13}$ units.

2. The distance between $B$ and $C$ must be exactly $\sqrt{52}$ units.

3. The three points must be collinear, satisfying the relationship $d(A,B) + d(B,C) = d(A,C)$.

Explain the reasoning you followed to construct their coordinates and analytically prove, by calculating the distance $d(A,C)$, that the points are indeed collinear without relying on a drawing.

A telecommunications company plans to install a repeater tower to provide coverage to two rural villages represented on a regional map by the points $A(-3, 2)$ and $B(5, 6)$ (coordinates measured in kilometres). For technical access reasons, the station must be built on a straight road whose route corresponds to the equation $y = 3x - 4$. Furthermore, to balance the received signal strength in both municipalities, the station $P(x, y)$ must be equidistant from both ($d(P,A) = d(P,B)$).

1. a) Model the equidistance condition by expanding the equality $d(P,A)^2 = d(P,B)^2$ and obtain the general linear equation of the perpendicular bisector of the segment connecting both villages.
2. b) Set up and solve the system of equations to find the exact coordinates of the antenna's location.
3. c) Calculate the exact distance in kilometres from the antenna to each of the two municipalities, checking that they coincide.