5.3 Coordinate Geometry, Area & Volume Mensuration

5.3 Coordinate Geometry, Area & Volume Mensuration

Coordinate geometry integrates algebra and visual geometry on a Cartesian plane, allowing geometric figures to be defined by numbers and algebraic equations. Mensuration focuses on quantifying lengths, perimeter, area, surface area, and volume of 2D and 3D shapes. Both topics are heavily featured in the HEC Aptitude Test (HAT).

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1. Coordinate Plane Analytics

Given two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ in the Cartesian coordinate plane:

Distance Formula

The distance $d$ between $P_1$ and $P_2$ is derived directly from the Pythagorean theorem:

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

Midpoint Formula

The midpoint $M$ of the line segment connecting $P_1$ and $P_2$ is the average of their respective coordinates:

$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Slope Formula

The slope $m$ measures the steepness and direction of a non-vertical line:

$m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \quad (x_1 \neq x_2)$

Forms of Line Equations

• Slope-Intercept Form: $y = mx + b$ (where $m$ is slope and $b$ is y-intercept).
• Point-Slope Form: $y - y_1 = m(x - x_1)$.
• Standard Form: $Ax + By = C$ (where $A, B, C$ are integers and $A \ge 0$).

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2. Parallel & Perpendicular Lines

Let line $L_1$ have slope $m_1$ and line $L_2$ have slope $m_2$:

• Parallel Lines ($L_1 \parallel L_2$): Lines never intersect and have identical slopes: $m_1 = m_2$
• Perpendicular Lines ($L_1 \perp L_2$): Lines intersect at $90^\circ$ angles, and their slopes are negative reciprocals: $m_1 \cdot m_2 = -1 \implies m_2 = -\frac{1}{m_1}$

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Worked Example 3.1: Perpendicular Line Equation

Problem: Find the equation of the line passing through point $A(4, -1)$ that is perpendicular to the line $3x - 2y = 8$.

Step-by-Step Derivation:

1. Rearrange $3x - 2y = 8$ into slope-intercept form to extract slope $m_1$: $-2y = -3x + 8 \implies y = \frac{3}{2}x - 4 \implies m_1 = \frac{3}{2}$
2. Determine perpendicular slope $m_2$: $m_2 = -\frac{1}{m_1} = -\frac{2}{3}$
3. Use point-slope form with $A(4, -1)$ and $m_2 = -2/3$: $y - (-1) = -\frac{2}{3}(x - 4) \implies y + 1 = -\frac{2}{3}x + \frac{8}{3}$
4. Subtract $1$ from both sides to convert to slope-intercept form: $y = -\frac{2}{3}x + \frac{8}{3} - \frac{3}{3} \implies y = -\frac{2}{3}x + \frac{5}{3}$

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3. Two-Dimensional (2D) Mensuration

Quadrilaterals & Polygons

Polygon Angle Properties

For any convex polygon with $n$ sides:

• Sum of Interior Angles: $S = (n - 2) \times 180^\circ$
• Each Interior Angle (Regular Polygon): $I = \frac{(n - 2) \times 180^\circ}{n}$
• Sum of Exterior Angles: Always $360^\circ$

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4. Three-Dimensional (3D) Mensuration

3D solids require computing Volume ($V$), Curved/Lateral Surface Area ($LSA$), and Total Surface Area ($TSA$).

Prisms & Cubes

• Rectangular Solid: $V = l \cdot w \cdot h$, $TSA = 2(lw + lh + wh)$. Space diagonal $D = \sqrt{l^2 + w^2 + h^2}$.
• Cube: $V = s^3$, $TSA = 6s^2$. Space diagonal $D = s\sqrt{3}$.

Circular Solids (Cylinders, Cones, Spheres)

$\text{Right Circular Cylinder: } V = \pi r^2 h, \quad LSA = 2\pi r h, \quad TSA = 2\pi r h + 2\pi r^2$

$\text{Right Circular Cone: } V = \frac{1}{3}\pi r^2 h, \quad \text{Slant height } l = \sqrt{r^2 + h^2}, \quad LSA = \pi r l$

$\text{Sphere: } V = \frac{4}{3}\pi r^3, \quad Surface Area = 4\pi r^2$

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Worked Example 3.2: Combined Solid Volume & Surface Area

Problem: A solid cylinder has a radius of $r = 3$ cm and a height of $h = 8$ cm. A cone of the same radius $3$ cm and height $4$ cm is attached to its flat top. Calculate the total combined volume of the composite solid.

Step-by-Step Derivation:

1. Calculate Volume of the Cylinder $V_{\text{cyl}}$: $V_{\text{cyl}} = \pi r^2 h = \pi (3^2)(8) = 72\pi \text{ cm}^3$
2. Calculate Volume of the Cone $V_{\text{cone}}$: $V_{\text{cone}} = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (3^2)(4) = \frac{1}{3}\pi (9)(4) = 12\pi \text{ cm}^3$
3. Total Volume $V_{\text{total}}$: $V_{\text{total}} = V_{\text{cyl}} + V_{\text{cone}} = 72\pi + 12\pi = 84\pi \text{ cm}^3$

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5. Scaling Laws for Geometric Solids

If all linear dimensions (length, width, height, radius) of a 3D figure are scaled by a linear factor $k$:

1. Any linear measure (perimeter, diagonal, circumference) changes by factor $k$.
2. Any surface area (base area, total surface area) changes by factor $k^2$.
3. Any volume changes by factor $k^3$.