Exercises: Distance between two points

In a test, students are asked to find the distance between points $A(-2, 1)$ and $B(1, 5)$. A student's response is $d(A, B) = \sqrt{(1 - (-2))^2 - (5 - 1)^2} = \sqrt{9 - 16} = \sqrt{-7}$, concluding that the distance does not exist. Identify the sign error in the formula used and calculate the correct distance.

Calculate the exact distance between points $A(3, -2)$ and $B(-1, 1)$ in the Cartesian plane.

Find the magnitude of vector $\vec{v} = (5, -12)$ using the corresponding formula.

A circle has its centre at point $C(-4, 2)$ and passes through point $P(0, 5)$. Calculate the exact length of its radius.

Determine how far point $M(-8, 15)$ is from the origin $O(0,0)$.

Calculate the perimetre of the triangle whose vertices are located at coordinates $A(0, 0)$, $B(6, 0)$ and $C(3, 4)$.

Check mathematically whether point $P(2, 3)$ is equidistant from points $M(5, -1)$ and $N(-1, -1)$.

Given the line segment with endpoints $R(-3, -4)$ and $S(2, 8)$, calculate the exact length of segment $\overline{RS}$.

In a Cartesian plane where each unit represents one metre, an ant moves in a straight line from position $P(-2, 7)$ to $Q(4, -1)$. Exactly how many metres has the ant travelled?

Marcos wants to calculate the straight-line distance between his secondary school, located at point $A(-3, 2)$, and the municipal sports centre, at point $B(5, -4)$, where each grid unit represents one hectometre ($1\text{ hm} = 100\text{ m}$). In his notebook, he writes the following:

1. a) Find the mistake Marcos made when substituting the coordinates into the formula and explain in detail why his reasoning is incorrect.
2. b) Correct his approach to calculate the actual, exact distance in hectometres (leaving the result in square root form) and calculate its approximate value rounded to two decimal places.

A forest fire surveillance drone from the Plan INFOCA in Andalusia takes off from an operations base located at point $A(2, 3)$ and travels in a straight line to a smoke column detected at coordinates $B(8, 11)$. On the command post grid, each unit represents $1\text{ km}$.

1. a) Model the flight path as the straight segment joining both points and calculate the exact distance in kilometres that the drone will travel to the fire.
2. b) If the drone flies at a constant cruising speed of $60\text{ km/h}$, how many minutes will it take exactly to reach point $B$? Explain why aerial emergency protocols use Euclidean distance instead of road network distance.

Find the value or possible values of the coordinate $a$ such that the distance between points $A(a, 3)$ and $B(2, -1)$ is exactly $5$ units.

Prove whether the triangle with vertices $A(-1, 2)$, $B(2, -2)$ and $C(6, 1)$ is a right-angled triangle by calculating the lengths of its three sides and applying the converse of Pythagoras' theorem.

A forest surveillance drone performs a patrol route flying in a straight line between three observation towers located at $T_1(-10, 5)$, $T_2(14, -2)$ and $T_3(2, 14)$, finally returning to $T_1$. If the coordinates are given in kilometres, calculate the total distance covered by the drone.

Find the coordinates of the point on the x-axis that is equidistant from the points $A(1, 4)$ and $B(5, -2)$.

Calculate the area of the triangle with vertices $P(0, -3)$, $Q(14, -3)$ and $R(5, 9)$ using Heron's formula after finding the exact length of its three sides.

A robotic arm needs to weld two components onto a circuit board. The first weld point is at $S_1(12, 5)$ and the second at $S_2(3, -7)$ (measurements in millimetres). If the arm head moves at a constant speed of $4 \text{ mm/s}$, how long will it take to travel in a straight line from $S_1$ to $S_2$?

The direction vector of a projectile's trajectory is given by the expression $\vec{u} = (k-1, k+1)$. Determine the possible values of the parameter $k$ so that the magnitude of this vector is $\sqrt{10}$ units.

A farmer wants to fence a circular orchard. He knows that the endpoints of one of the diameters of the orchard are located at coordinates $M(-3, -5)$ and $N(7, 1)$ on his topographic map (where $1 \text{ unidad} = 10 \text{ m}$). Calculate the total area of the orchard in square metres.

The AEMET weather radar detects a storm core at point $T(9, 8)$ that is moving towards a regional station located at point $E(2, 3)$, where each map grid unit is equivalent to $10\text{ km}$.

1. a) Before calculating the square root, make a reasoned estimate: calculate the horizontal separation $|x_2 - x_1|$ and vertical separation $|y_2 - y_1|$ in kilometres. Based on these two legs, argue between which two integer values of tens of kilometres (multiples of $10$) the actual straight-line distance must necessarily lie.
2. b) Apply the analytical distance formula to find the exact value in kilometres and round the result to the nearest tenth. Check whether the value obtained fits within your previous estimate and explain why the hypotenuse of this right triangle is always greater than the longer leg but strictly less than the sum of both legs.

A telecommunications junction box is located at point $A(2, 5)$ on a technical drawing. The plan is to connect it via a straight underground service pipe to a fixed horizontal duct that runs exactly along the line $y = 1$. The length of cable available for the section is $5\text{ m}$ (each unit on the drawing equals $1\text{ m}$).

1. a) Write a quadratic equation in the unknown $x$ that represents the mathematical condition for a generic point $P(x, 1)$ on the duct to be exactly $5\text{ m}$ away from $A$.
2. b) Solve this equation to find the coordinates of the two possible points where the cable can be connected, and explain geometrically why two solutions symmetrical with respect to the position of the junction box are obtained.

A comet is at position $C(10, 15)$ and a space probe follows a linear trajectory given by $2x - y - 1 = 0$. A novice scientist chooses a random point on the trajectory, for example $P(1, 1)$, and calculates the distance from the comet to the trajectory as $d(C, P) = \sqrt{(10-1)^2 + (15-1)^2} = \sqrt{81 + 196} = \sqrt{277}$. Explain why this method does not guarantee finding the actual distance to the line. Apply the correct analytical formula and find the exact value of this distance.

Two of the vertices of an equilateral triangle are located at coordinates $A(-2, 0)$ and $B(2, 0)$. Determine the possible coordinates of the third vertex $C(x, y)$, knowing that the distance between all its vertices is identical.

A telecommunications antenna is to be installed to provide coverage with equal intensity to three villages located at coordinates $P_1(-3, 2)$, $P_2(5, -4)$ and $P_3(4, 3)$. Determine the exact coordinates of the circumcentre where the antenna should be placed so that it is equidistant from the three villages.

Given the quadrilateral with vertices $A(-4, 1)$, $B(2, -1)$, $C(4, 5)$ and $D(-2, 7)$, verify that it is a perfect square by calculating the lengths of its four sides and its two diagonals.

Find the exact coordinates of the point $P$ that lies on the line with equation $y = 2x - 3$ and is equidistant from the points $A(-1, 4)$ and $B(5, 2)$.

In the design of a solar roof, a panel is supported by two vertical posts. In a cross-sectional plane, the base of the short post is at $O(0,0)$ and that of the tall post is at $B(6,0)$. The top end of the short post is at $A(0, 2)$ and that of the tall post is at $C(6, 5)$. A guy wire must go from $A$ to an anchor point $P(x, 0)$ on the ground, and from there go up to $C$. Determine the exact value of $x$ that minimises the total length of the wire $d(A,P) + d(P,C)$.

The points $A(-1, 2)$ and $B(3, k)$ define a line segment whose length is exactly double the length of the line segment formed by the points $C(0, 0)$ and $D(1, -2)$. Set up the corresponding equation and calculate the possible values of the parameter $k$.

A comet follows a straight-line path that, on a star map, passes through coordinates $C_1(-2, -2)$ and $C_2(6, 6)$ expressed in astronomical units (AU). An observation station is at the fixed position $P(5, 1)$. Calculate the minimum distance at which the comet will pass the observation station.

The endpoints of one of the diagonals of a square are located at points $M(-2, 5)$ and $N(6, -1)$. Knowing that the diagonals of a square intersect at their midpoint, are perpendicular, and have exactly the same length, find the coordinates of the other two vertices $P$ and $Q$, and calculate the exact perimeter of the square.

A solar plant is planned to be installed with its control centre at point $P(5, 7)$. To export the power generated, a connection line must be run to an existing electrical infrastructure, whose layout follows the line with general equation $3x - 4y + 1 = 0$ (with units in hundreds of metres). Two technicians discuss the method for calculating the minimum cable length:

• Eva's method: Use the distance from a point to a line formula: $d(P, r) = \dfrac{|A \cdot p_1 + B \cdot p_2 + C|}{\sqrt{A^2 + B^2}}$.

• Iván's method: Choose any point on the power line, such as $Q(1, 1)$, and calculate the distance between two points using $d(P, Q) = \sqrt{(1 - 5)^2 + (1 - 7)^2}$.

1. a) Apply Eva's method and determine the exact minimum length that the transmission cable must have.
2. b) Explain why the method proposed by Iván does not solve the problem posed and would lead to an unnecessary extra cost when purchasing cable.
3. c) If you did not remember Eva's direct formula, describe step by step what geometric procedure you should follow to find the correct distance using only the distance formula between two points.

The Consorcio Provincial de Bomberos de Toledo wants to locate an emergency helipad to provide fast and equitable service to two mountain municipalities located at $A(2, 1)$ and $B(8, 5)$, on a regional map where each cartographic unit represents $4\text{ km}$. For logistics reasons regarding refuelling via tanker truck, the helipad $H(x, y)$ must necessarily be built on a straight regional road defined by the equation $y = 6$.

1. a) Model the equidistance condition by setting up the equation $d(H, A) = d(H, B)$ for a generic point on the road $H(x, 6)$ and solve algebraically to find the exact cartographic coordinates of the helipad.
2. b) Calculate the actual straight-line distance (in kilometres, rounded to the nearest tenth) from the helipad to either of the two municipalities served.
3. c) The mayor's office of one of the municipalities proposes that the helipad be built at the midpoint of segment $AB$. Find the coordinates of that midpoint and justify, using mathematical and spatial arguments, why that local proposal is not viable for this project.