Exercises: Distance between two points

Calculate the exact distance between the points $A(2, 3)$ and $B(5, 7)$ on the Cartesian plane.

A ship is located at the coordinates $P(-4, 1)$ and a lighthouse at $Q(2, -7)$. If each unit on the plane is equivalent to one mile, at what straight-line distance is the ship from the lighthouse?

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

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 a two-dimensional radar screen, two airplanes are located at the points $R(-3, -2)$ and $S(9, 3)$. Determine the linear distance between them in radar units.

Calculate the length of the diagonal of a rectangle if you know that two of its opposite vertices have the coordinates $V_1(-1, 4)$ and $V_2(5, -4)$.

Check mathematically if the segments $\overline{AB}$ and $\overline{CD}$ have exactly the same length, given the coordinates of the points $A(0, 2)$, $B(3, 6)$, $C(-2, -1)$ and $D(2, 2)$.

A surveyor marks two points on flat ground: $T_1(12, 5)$ and $T_2(24, 10)$. If the coordinates are expressed in feet, how many feet of cable does he need to connect them in a straight line?

The distance between the points $A(1, 2)$ and $B(x, 5)$ is exactly $5$ units. Find the two possible values of the coordinate $x$.

A delivery drone takes off from the logistics center $O(0,0)$, flies in a straight line to the first delivery point $A(6, 8)$ and then changes course to head to a second point at $B(15, -4)$. If each unit on the plane represents $100$ feet, what is the total distance traveled by the drone in miles?

On an architectural plan, three main columns will be located at the coordinates $P(-2, 1)$, $Q(2, 4)$ and $R(5, 0)$. Prove mathematically whether the triangle formed by these three columns is isosceles.

A telecommunications antenna has a maximum circular range of $10\text{ km}$. If the antenna is located at the point $C(3, -2)$ on a map (where $1$ coordinate unit is equivalent to $1\text{ km}$), find out if a user located at the position $U(-4, 5)$ is within the coverage area.

Find the coordinates of all points located on the x-axis that are exactly $10$ units of distance from the point $M(-6, 8)$.

In a set-piece soccer play, a player located at $J_1(10, 20)$ makes a low pass to his teammate at $J_2(34, 27)$, who without stopping the ball shoots directly toward the center of the goal at $P(50, 15)$. Calculate the total distance the ball travels. (The coordinates are given in feet).

Prove, by calculating the lengths of the three sides and using the Pythagorean theorem, that the triangle with vertices $A(2, 1)$, $B(6, 1)$ and $C(2, 4)$ is a right triangle.

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

In a 3D space flight simulator, a spacecraft moves from the coordinate $P_1(2, -1, 4)$ to the orbital station at $P_2(5, 3, 16)$. Calculate the exact linear distance the spacecraft travels in space.

A straight pipe must connect a well located at the point $A(-3, 2)$ with a water pump $P$ that must necessarily be installed on the y-axis. If due to pressure restrictions the length of the pipe must be exactly $\sqrt{29}$ units, find the possible coordinates for the pump $P$.

The vertices of a triangular-shaped plot of land are $A(-4, -2)$, $B(2, 6)$ and $C(6, 3)$. First prove that the plot forms a right triangle and, then, calculate its perimeter and its exact area.

An automated irrigation system consists of a circular sprinkler with its center at $C(x, y)$ that must equally water three trees located at the coordinates $T_1(0, 0)$, $T_2(6, 0)$ and $T_3(0, 8)$. Determine the coordinates where the sprinkler $C$ must be placed and what its exact radius of reach will be.

Determine whether the points $A(-2, 1)$, $B(1, 3)$ and $C(7, 7)$ are perfectly collinear. To do this, calculate the distances $d(A,B)$, $d(B,C)$ and $d(A,C)$, and check if the equation $d(A,B) + d(B,C) = d(A,C)$ holds.

A telecommunications company wants to install a fiber optic cable connecting two buildings $E_1(0, 0)$ and $E_2(21, 0)$. The installation cost is $150\text{ \$}$ per unit of length. If they decide to add an intermediate node at $N(16, 12)$ and connect $E_1$ to $N$, and then $N$ to $E_2$, how much extra money will this deviation cost compared to the direct connection between the two buildings?