Distance Formula in Two Dimensions

Derivation of the Distance Formula

The distance formula is a fundamental tool in coordinate geometry used to calculate the length of the straight line segment connecting two points in a Cartesian plane. Its derivation relies directly on the application of the Pythagorean theorem.

Let's consider two distinct points in the two-dimensional Cartesian plane:

• Point P with coordinates $(x_1, y_1)$
• Point Q with coordinates $(x_2, y_2)$

Our objective is to find the distance, denoted by $d$, between these two points P and Q. This distance is represented by the length of the line segment $\overline{PQ}$.

Derivation using Pythagorean Theorem

To use the Pythagorean theorem, we need to construct a right-angled triangle involving the points P and Q. We can do this by drawing lines parallel to the coordinate axes.

Construction:

1. Draw a line through point P that is parallel to the x-axis.
2. Draw a line through point Q that is parallel to the y-axis.
3. Let these two constructed lines intersect at a point, which we will call R.

Now, let's determine the coordinates of this intersection point R. Since the line segment PR is parallel to the x-axis, any point on this line has the same y-coordinate as P. Thus, the y-coordinate of R is $y_1$. Since the line segment QR is parallel to the y-axis, any point on this line has the same x-coordinate as Q. Thus, the x-coordinate of R is $x_2$. Therefore, the coordinates of point R are $(x_2, y_1)$.

Consider the triangle formed by the points P, R, and Q, which is $\triangle PRQ$.

• The side PR is a horizontal line segment because it is parallel to the x-axis.
• The side QR is a vertical line segment because it is parallel to the y-axis.

Since the x-axis and y-axis are perpendicular to each other, any line segment parallel to the x-axis is perpendicular to any line segment parallel to the y-axis. Therefore, the angle at R, $\angle PRQ$, is $90^\circ$. This confirms that $\triangle PRQ$ is a right-angled triangle with PQ as its hypotenuse.

Calculating the Lengths of the Sides

Now we calculate the lengths of the two shorter sides, PR and QR, using the coordinates of the points:

• The length of the horizontal side PR is the absolute difference between the x-coordinates of its endpoints P$(x_1, y_1)$ and R$(x_2, y_1)$. The y-coordinates are the same, which is characteristic of a horizontal line segment.

  Length PR = $|x_2 - x_1|$

• The length of the vertical side QR is the absolute difference between the y-coordinates of its endpoints Q$(x_2, y_2)$ and R$(x_2, y_1)$. The x-coordinates are the same, which is characteristic of a vertical line segment.

  Length QR = $|y_2 - y_1|$

• The length of the hypotenuse PQ is the distance $d$ we want to find.

  Length PQ = $d$

Applying the Pythagorean Theorem

According to the Pythagorean theorem, in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

In $\triangle PRQ$, we have:

Substitute the lengths we calculated:

We know that for any real number $a$, $(|a|)^2 = a^2$. This is because squaring a number always results in a non-negative value, and the absolute value only affects the sign, which is eliminated by squaring.

Applying this property to equation (ii):

To find the distance $d$, we take the square root of both sides of equation (iii). Since distance is a non-negative quantity, we take the positive square root:

This is the standard form of the distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ in a two-dimensional Cartesian plane.

Important Note on Order of Subtraction

Since $(x_2 - x_1)^2 = (x_1 - x_2)^2$ (squaring a negative number gives the same result as squaring its positive counterpart) and similarly $(y_2 - y_1)^2 = (y_1 - y_2)^2$, the order in which you subtract the coordinates does not affect the final distance. For example, the distance between $(1, 2)$ and $(4, 6)$ can be calculated as:

$d = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

or

$d = \sqrt{(1 - 4)^2 + (2 - 6)^2} = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

The result is the same.

Distance from the Origin

A special case of the distance formula is finding the distance of a point P$(x, y)$ from the origin O$(0, 0)$. Using the distance formula with $(x_1, y_1) = (0, 0)$ and $(x_2, y_2) = (x, y)$, we get:

$d = \sqrt{(x - 0)^2 + (y - 0)^2}$

$d = \sqrt{x^2 + y^2}$

So, the distance of a point $(x, y)$ from the origin is $\sqrt{x^2 + y^2}$.

Calculating Distance Between Two Points

The distance formula is a direct application of the Pythagorean theorem, allowing us to calculate the shortest distance between any two points in a two-dimensional Cartesian coordinate system. Given two points $P(x_1, y_1)$ and $Q(x_2, y_2)$, the distance $d$ between them is given by the formula:

$\mathbf{d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}$

This formula measures the length of the straight line segment connecting P and Q.

Steps to Calculate Distance:

To calculate the distance between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ using the distance formula, follow these steps:

1. Identify Coordinates: Clearly identify the coordinates of the two points. Let the first point be $(x_1, y_1)$ and the second point be $(x_2, y_2)$. The choice of which point is $(x_1, y_1)$ and which is $(x_2, y_2)$ does not affect the final distance, as $(x_2 - x_1)^2 = (x_1 - x_2)^2$ and $(y_2 - y_1)^2 = (y_1 - y_2)^2$.

2. Find the Difference in x-coordinates: Subtract the x-coordinate of the first point from the x-coordinate of the second point: $(x_2 - x_1)$.

3. Square the Difference in x-coordinates: Square the result from Step 2: $(x_2 - x_1)^2$.

4. Find the Difference in y-coordinates: Subtract the y-coordinate of the first point from the y-coordinate of the second point: $(y_2 - y_1)$.

5. Square the Difference in y-coordinates: Square the result from Step 4: $(y_2 - y_1)^2$.

6. Sum the Squared Differences: Add the squared differences from Step 3 and Step 5: $(x_2 - x_1)^2 + (y_2 - y_1)^2$.

7. Take the Square Root: Calculate the positive square root of the sum obtained in Step 6. This final value is the distance $d$ between the two points.

   $d = \sqrt{\text{Sum from Step 6}}$

Answer:

Answer:

Distance from the Origin

A frequently encountered scenario is finding the distance of a point from the origin O(0, 0). We can derive a simplified formula for this specific case using the general distance formula. Consider a point $P(x, y)$ and the origin $O(0, 0)$. Let $(x_1, y_1) = (0, 0)$ and $(x_2, y_2) = (x, y)$.

Using the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Substitute the coordinates of P and O:

$d = \sqrt{(x - 0)^2 + (y - 0)^2}$

$d = \sqrt{x^2 + y^2}$

So, the distance of a point $(x, y)$ from the origin $(0, 0)$ is given by:

$\mathbf{d = \sqrt{x^2 + y^2}}$

Answer:

Applications of Distance Formula (e.g., verifying geometric properties, collinearity)

The distance formula, derived from the Pythagorean theorem, is a powerful analytical tool in coordinate geometry. It allows us to measure the lengths of segments connecting points, which in turn enables us to verify fundamental geometric properties, classify figures, and solve various problems involving distances between points or sets of points.

Given the distance formula between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ as $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$, let's explore its primary applications.

1. Verifying Properties of Geometric Shapes

By calculating the lengths of sides, diagonals, and other segments using the distance formula, we can determine the type of geometric figure formed by a given set of vertices. This method provides a rigorous way to prove properties that might otherwise only be observed visually from a graph.

Properties of Triangles:

Given the vertices of a triangle, say A, B, and C, we calculate the lengths of the three sides AB, BC, and CA using the distance formula. Based on these lengths, we can classify the triangle:

• Scalene Triangle: If all three sides have different lengths (i.e., $AB \neq BC$, $BC \neq CA$, and $CA \neq AB$), the triangle is scalene.
• Isosceles Triangle: If any two sides have equal lengths (e.g., $AB = BC$, or $BC = CA$, or $CA = AB$), the triangle is isosceles.
• Equilateral Triangle: If all three sides have equal lengths (i.e., $AB = BC = CA$), the triangle is equilateral. An equilateral triangle is a special case of an isosceles triangle.
• Right-Angled Triangle: To check if a triangle is right-angled, we use the converse of the Pythagorean theorem. Calculate the squares of the lengths of all three sides ($AB^2$, $BC^2$, $CA^2$). If the square of the longest side is equal to the sum of the squares of the other two sides, then the triangle is right-angled. For example, if $AC$ is the longest side, check if $AB^2 + BC^2 = AC^2$.

Properties of Quadrilaterals:

Given the vertices of a quadrilateral, say A, B, C, and D, we calculate the lengths of the four sides (AB, BC, CD, DA) and the two diagonals (AC, BD). Based on these lengths, we can classify the quadrilateral:

• Parallelogram: Opposite sides are equal in length ($AB = CD$ and $BC = DA$).
• Rectangle: A parallelogram with equal diagonals ($AB = CD$, $BC = DA$, and $AC = BD$). Alternatively, a rectangle is a quadrilateral where opposite sides are equal and adjacent sides are perpendicular (which can be checked using slopes, or by verifying if the Pythagorean theorem holds for the triangles formed by adjacent sides and a diagonal). Using only distance, check if opposite sides are equal and diagonals are equal.
• Rhombus: All four sides are equal in length ($AB = BC = CD = DA$).
• Square: A rhombus with equal diagonals ($AB = BC = CD = DA$ and $AC = BD$). Alternatively, a square is a rectangle with all sides equal. Using only distance, check if all four sides are equal AND the diagonals are equal.
• Trapezium (or Trapezoid): At least one pair of opposite sides is parallel. The distance formula alone cannot determine parallelism; slopes are needed for this. However, for an Isosceles Trapezium, the non-parallel sides are equal in length.

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2. Checking for Collinearity of Three Points

Three or more points are said to be collinear if they lie on the same straight line. The distance formula provides a method to test for collinearity without using slopes or the concept of the area of a triangle.

To check if three distinct points A, B, and C are collinear using the distance formula, we calculate the distances between all three pairs of points: $d(A, B)$, $d(B, C)$, and $d(A, C)$.

If the points are collinear, then the sum of the lengths of the two shorter segments must be equal to the length of the longest segment. For example, if B lies between A and C on the line, then the distance from A to B plus the distance from B to C must equal the distance from A to C. That is, $AB + BC = AC$.

Condition for Collinearity: Three points A, B, and C are collinear if and only if one of the following three conditions holds true:

• $d(A, B) + d(B, C) = d(A, C)$
• $d(A, C) + d(C, B) = d(A, B)$
• $d(B, A) + d(A, C) = d(B, C)$

In simpler terms, pick the two distances that sum up to the largest distance.

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3. Finding Points Equidistant from Others

A common application is finding the coordinates of a point that is located at the same distance from two or more given points. This is useful in various geometric constructions and problems.

• The set of all points equidistant from two given points forms the perpendicular bisector of the line segment joining the two points.
• The circumcenter of a triangle is the point equidistant from all three vertices of the triangle.

To find a point P that is equidistant from points A and B, we set the distance $d(P, A)$ equal to the distance $d(P, B)$, i.e., $PA = PB$. Squaring both sides ($PA^2 = PB^2$) often simplifies the calculation by removing the square roots.

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