The Distance Formula
In a video game map, your character stands at $(1, 2)$ and a treasure chest sits at $(13, 7)$. How far is the chest in a straight line? GPS apps, radar screens and game engines answer this kind of question constantly, and on the coordinate plane they always use the same tool: the Pythagorean Theorem.
The Distance Formula
Pick two points $A(x_1, y_1)$ and $B(x_2, y_2)$. Draw a horizontal segment from $A$ and a vertical segment from $B$ until they meet. You get a right triangle whose legs are the horizontal change $x_2 - x_1$ and the vertical change $y_2 - y_1$. The segment $AB$ is the hypotenuse.
By the Pythagorean Theorem, $d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2$. Take the square root and you have the formula.
The order of the points does not matter. $(x_2 - x_1)^2$ and $(x_1 - x_2)^2$ are equal because squaring erases the sign. This is why the distance from $A$ to $B$ always equals the distance from $B$ to $A$. If two segments have equal distances, they have the same length.
Calculate the exact distance between the points $A(2, 3)$ and $B(5, 7)$ on the Cartesian plane.
Calculate the exact distance between the points $A(2, 3)$ and $B(5, 7)$ on the Cartesian plane.
Find the length of the segment whose endpoints are the origin $O(0, 0)$ and the point $M(-8, 6)$.
Find the length of the segment whose endpoints are the origin $O(0, 0)$ and the point $M(-8, 6)$.
Check mathematically if the segments $\overline{AB}$ and $\overline{CD}$ have exactly the same length, given the coordinates of the points $A(0, 2)$, $B(3, 6)$,…
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 $D(2, 2)$.
Perimeter and Types of Triangles
With the distance formula you can find the perimeter of any polygon by adding the lengths of its sides. You can also classify a triangle without drawing it to scale.
Why compare squares? $d^2$ is just the sum under the radical. It is exact, has no roots, and is easier to compare. If $d_1^2 = d_2^2$, then $d_1 = d_2$.
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 located at the 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 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.
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…
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.
Midpoints and Parallelograms
Some problems ask for the point exactly halfway between two points. To find it, average the coordinates.
The midpoint is halfway along each axis, so it is halfway along the segment. That means $AM = MB = \tfrac{1}{2}AB$.
Parallelograms. The diagonals of a parallelogram bisect each other, so they share the same midpoint. This lets you find a missing vertex.
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…
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.
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…
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.
Finding Points at a Given Distance
Sometimes the distance is given and a coordinate is unknown. Write the distance formula with the unknown in it, square both sides to remove the root, and solve.
All the points at distance $13$ from $M$ form a circle. A point on the $x$-axis is a point where the circle crosses the axis, so you can expect up to two solutions.
When the distance is a radical, such as $\sqrt{29}$, squaring turns it into a plain number ($29$). You may end up with an equation of the form $x^2 = k$, or with a quadratic that you solve by factoring.
The distance between the points $A(1, 2)$ and $B(x, 5)$ is exactly $5$ units. Find the two possible values of the coordinate $x$.
The distance between the points $A(1, 2)$ and $B(x, 5)$ is exactly $5$ units. Find the two possible values of the coordinate $x$.
Find the coordinates of all points located on the x-axis that are exactly $10$ units of distance from the point $M(-6, 8)$.
Find the coordinates of all points located on the x-axis that are exactly $10$ units of distance from the point $M(-6, 8)$.
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…
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 $P$.
Collinear Points
Three points are collinear if they lie on one line. Distance gives a test for this.
Why? In any real triangle, two sides together are longer than the third (the Triangle Inequality). The only way two sides can add up to exactly the third is if the triangle is flattened into a straight segment.
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…
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.
Equidistant Points and Circles
A point $P$ is equidistant from $A$ and $B$ if $PA = PB$. Since distances are never negative, you can square both sides and drop the roots. Even better, the $x^2$ and $y^2$ terms often cancel and leave a simple linear equation.
All points equidistant from $A$ and $B$ form the perpendicular bisector of $\overline{AB}$.
Three points. A point equidistant from three points is the center of the circle through all three (the circumcenter of the triangle). Write two equations ($PA = PB$ and $PA = PC$), solve for $P$, and the radius is the distance from $P$ to any vertex.
Inside or outside? Compare the distance from a point to the center with the radius $r$. If $d < r$ the point is inside the circle, if $d = r$ it is on the circle, and if $d > r$ it is outside.
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…
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 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…
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 $B(0, 4)$.
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 $T_1(0, 0)$,…
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.
Distance in 3D Space
In space, a point has three coordinates $(x, y, z)$. The idea is the same, with a third change added. First use Pythagoras on the floor of an imaginary box, $\sqrt{\Delta x^2 + \Delta y^2}$. Then use it again with the vertical change $\Delta z$.
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…
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.
Distance in the Real World
In word problems, work in coordinate units first, then use the scale to convert to feet or miles. Remember that $1\text{ mi} = 5{,}280\text{ ft}$.
Paths with several legs mean several distances. Add them for the total distance traveled. A detour costs the difference between the long route and the direct route.
Back to the opening. The character at $(1, 2)$ and the chest at $(13, 7)$ have changes $12$ and $5$, so the straight-line distance is $\sqrt{12^2 + 5^2} = \sqrt{169} = 13$ units. If one unit is $10$ ft, the chest is $130$ ft away.
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…
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?
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…
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).
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{ \$}…
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 $E1$ to $N$, and then $N$ to $E2$, how much extra money will this deviation cost compared to the direct connection between the two buildings?