8.4 The Coordinate Plane, Pythagorean Theorem & Distance

8.4 The Coordinate Plane, Pythagorean Theorem & Distance

Quick Answer: The Cartesian coordinate plane is formed by perpendicular $x$-axis and $y$-axis lines intersecting at the origin $(0,0)$, partitioning space into four quadrants where signs indicate displacement directions. Geometric transformations include rigid motions (isometries: translations, reflections, rotations) that preserve side lengths, angle measures, and congruence, and non-rigid motions (dilations by scale factor $k$) that preserve shape and angle measures to yield similar figures. In any right triangle, the Pythagorean Theorem establishes $a^2 + b^2 = c^2$, with its converse confirming a right triangle whenever this equality holds. On the coordinate plane, the distance between any two points $(x_1, y_1)$ and $(x_2, y_2)$ is derived directly from the Pythagorean theorem: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

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Structure of the Cartesian Coordinate Plane

The Cartesian coordinate plane is constructed by two perpendicular real number lines intersecting at their respective zero points:

1. $x$-axis: The horizontal coordinate line, with positive values directed to the right and negative values directed to the left.
2. $y$-axis: The vertical coordinate line, with positive values directed upward and negative values directed downward.
3. Origin $(0,0)$: The central reference point of intersection.

The Four Quadrants

The intersecting axes divide the infinite two-dimensional plane into four regions called quadrants, designated by Roman numerals in a standard counterclockwise sequence starting from the upper right:

$$\begin{array}{c|c} \textbf{Quadrant II} & \textbf{Quadrant I} \\ (-, +) & (+, +) \\ x < 0, \; y > 0 & x > 0, \; y > 0 \\ \hline \textbf{Quadrant III} & \textbf{Quadrant IV} \\ (-, -) & (+, -) \\ x < 0, \; y < 0 & x > 0, \; y < 0 \end{array}$$

• Points on Axes: Any point situated directly on a coordinate axis does not belong to any quadrant. Points on the $x$-axis have a $y$-coordinate of zero ($(x, 0)$), while points on the $y$-axis have an $x$-coordinate of zero ($(0, y)$).

Grid Distance Between Aligned Points

When two points lie along a common horizontal or vertical grid line, distance is computed using absolute differences:

• Horizontal Segment (identical $y$-coordinates): The distance between $(x_1, y)$ and $(x_2, y)$ is: $$d = |x_2 - x_1|$$
• Vertical Segment (identical $x$-coordinates): The distance between $(x, y_1)$ and $(x, y_2)$ is: $$d = |y_2 - y_1|$$

Worked Example: A rectangle plotted on the coordinate plane has vertices at $A(-4, 3)$, $B(5, 3)$, $C(5, -2)$, and $D(-4, -2)$. What are its perimeter and area?

• Length (horizontal segments $AB$ and $DC$): $|5 - (-4)| = |9| = 9\text{ units}$.
• Width (vertical segments $BC$ and $AD$): $|3 - (-2)| = |5| = 5\text{ units}$.
• Perimeter: $P = 2(9) + 2(5) = 18 + 10 = 28\text{ units}$.
• Area: $A = 9 \times 5 = 45\text{ units}^2$.

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The Pythagorean Theorem & Its Converse

In any right triangle, the two sides that form the $90^\circ$ angle are called the legs (conventionally denoted $a$ and $b$), and the side directly opposite the right angle is the hypotenuse (denoted $c$). The hypotenuse is strictly the longest side of a right triangle.

The Pythagorean Theorem

In every right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse:

$$a^2 + b^2 = c^2$$

• Solving for the Hypotenuse: $$c = \sqrt{a^2 + b^2}$$
• Solving for an Unknown Leg: $$a = \sqrt{c^2 - b^2} \qquad b = \sqrt{c^2 - a^2}$$

Common Primitive Pythagorean Triples

A Pythagorean triple consists of three positive integers $(a, b, c)$ satisfying $a^2 + b^2 = c^2$:

• $(3, 4, 5)$ and its scalar multiples $(6, 8, 10)$, $(9, 12, 15)$, $(12, 16, 20)$
• $(5, 12, 13)$ and its scalar multiples $(10, 24, 26)$
• $(8, 15, 17)$
• $(7, 24, 25)$
• $(9, 40, 41)$

The Converse of the Pythagorean Theorem

The Converse of the Pythagorean Theorem states that if the side lengths of a triangle $a$, $b$, and $c$ (where $c$ is the longest side) satisfy $a^2 + b^2 = c^2$, then the triangle is guaranteed to be a right triangle.

Furthermore, comparing $a^2 + b^2$ with $c^2$ allows the classification of any valid triangle:

• If $a^2 + b^2 = c^2$, the triangle is a Right Triangle ($m\angle C = 90^\circ$).
• If $a^2 + b^2 > c^2$, the triangle is an Acute Triangle ($m\angle C < 90^\circ$).
• If $a^2 + b^2 < c^2$, the triangle is an Obtuse Triangle ($90^\circ < m\angle C < 180^\circ$).

Worked Example: Classify a triangle with side lengths $7\text{ cm}$, $24\text{ cm}$, and $25\text{ cm}$:

$$a^2 + b^2 = 7^2 + 24^2 = 49 + 576 = 625$$ $$c^2 = 25^2 = 625$$

Because $49 + 576 = 625$ ($a^2 + b^2 = c^2$), the triangle is confirmed to be a right triangle.

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The Coordinate Distance Formula

To determine the straight-line Euclidean distance $d$ between any two arbitrary points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ on the Cartesian coordinate plane, construct a right triangle whose legs lie parallel to the coordinate axes:

• The horizontal leg has length $\Delta x = |x_2 - x_1|$.
• The vertical leg has length $\Delta y = |y_2 - y_1|$.

Applying the Pythagorean Theorem ($(\Delta x)^2 + (\Delta y)^2 = d^2$) establishes the Euclidean Distance Formula:

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

Because squaring any real number eliminates negative signs ($(-a)^2 = a^2$), the direction of subtraction within each squared term does not alter the result: $(x_2 - x_1)^2 = (x_1 - x_2)^2$.

Step-by-Step Worked Example

Problem: Find the distance between point $A(-5, 7)$ and point $B(3, -8)$ on the coordinate plane.

1. Identify Coordinates: $(x_1, y_1) = (-5, 7)$ and $(x_2, y_2) = (3, -8)$.
2. Calculate Displacements: $$\Delta x = x_2 - x_1 = 3 - (-5) = 3 + 5 = 8$$ $$\Delta y = y_2 - y_1 = -8 - 7 = -15$$
3. Square the Displacements: $$(\Delta x)^2 = 8^2 = 64$$ $$(\Delta y)^2 = (-15)^2 = 225$$
4. Sum and Extract Square Root: $$d = \sqrt{64 + 225} = \sqrt{289} = 17\text{ units}$$

The distance between $A$ and $B$ is exactly $17$ units, forming an $(8, 15, 17)$ right triangle.

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Common Exam Traps & Misconceptions