3.4 Geometry, Measurement, and Data Interpretation

3.4 Geometry, Measurement, and Data Interpretation

The final module of Numerical Reasoning synthesizes spatial-quantitative measurement with data literacy. Geometry items evaluate your grasp of plane figures, geometric scaling, angle theorems, and solids, while data interpretation tests your capacity to extract facts, discern trends, and compute summary statistics from graphs and tables under timed conditions.

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Plane Geometry: Perimeter and Area

Perimeter is the one-dimensional total distance around the boundary of a closed figure, whereas area measures the two-dimensional surface enclosed within that boundary.

Primary Plane Figures and Area Formulations

Special Right Triangles

Memorizing the exact side ratios of special right triangles eliminates the need for manual square root extraction:

• $45^\circ - 45^\circ - 90^\circ$ Triangle (Isosceles Right): Side ratio is $1 : 1 : \sqrt{2}$. If legs are length $x$, the hypotenuse is $x\sqrt{2}$.
• $30^\circ - 60^\circ - 90^\circ$ Triangle: Side ratio is $1 : \sqrt{3} : 2$. Opposite the $30^\circ$ angle is $x$ (shortest leg), opposite $60^\circ$ is $x\sqrt{3}$, and opposite $90^\circ$ is $2x$ (hypotenuse).

Composite Plane Figures

To compute the area of composite figures (e.g., an L-shaped room, an athletic track, or a shaded border), decompose the figure into standard shapes. Either:

1. Partition into non-overlapping sub-rectangles/triangles and sum their individual areas; or
2. Compute the bounding area and subtract the unshaded cutout portion ($A_{\text{shaded}} = A_{\text{outer}} - A_{\text{inner}}$).

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Solid Geometry: Volume and Geometric Scaling Laws

Volume quantifies three-dimensional capacity, measured in cubic units ($m^3$, $cm^3$, or liters, where $1\text{ liter} = 1,000\text{ cm}^3$).

Standard 3D Solids

• Rectangular Prism (Box): $V = l \times w \times h$ $\text{Total Surface Area } (TSA) = 2(lw + lh + wh)$ $\text{Space Diagonal} = \sqrt{l^2 + w^2 + h^2}$

• Right Circular Cylinder: $V = \pi r^2 h$ $\text{Lateral Surface Area } (LSA) = 2\pi r h$ $\text{Total Surface Area } (TSA) = 2\pi r h + 2\pi r^2$

The Fundamental Geometric Scaling Laws

Questions frequently test what occurs when dimensions are scaled by a constant factor $k$:

• All linear dimensions (perimeter, circumference, radius, height, diagonal) scale by $k^1$.
• All surface areas (lateral area, base area, total surface area) scale by $k^2$.
• All volumes and capacities scale by $k^3$.

Non-Calculator Example: If the radius of a cylindrical water tank is doubled while its height remains constant, how does its volume change? The radius is squared in the volume formula ($V = \pi r^2 h$), so doubling the radius ($k = 2$) multiplies the volume by $2^2 = 4$.

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Angle Theorems and the Pythagorean Theorem

Angle Fundamentals

• Complementary Angles: Two angles whose measures sum to $90^\circ$.
• Supplementary Angles: Two angles whose measures sum to $180^\circ$.
• Vertical Angles: Non-adjacent angles formed by two intersecting lines; vertical angles are always congruent (equal in measure).
• Triangle Angle Sum Theorem: The interior angles of any planar triangle sum to $180^\circ$.
• Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.

Parallel Lines Cut by a Transversal

When two parallel lines are intersected by a transversal line, eight angles are created, forming exactly two equal measure groups:

• Congruent pairs: Alternate interior angles, alternate exterior angles, and corresponding angles are equal.
• Supplementary pairs: Consecutive interior angles (co-interior) on the same side of the transversal sum to $180^\circ$.

The Pythagorean Theorem and Primitive Triples

In every right triangle with legs $a$ and $b$ and hypotenuse $c$: $a^2 + b^2 = c^2$

To save time on the non-calculator exam, memorize the essential primitive Pythagorean triples and their common scalar multiples ($k \times \{a, b, c\}$):

• Family $\{3, 4, 5\}$: Multiples include $\{6, 8, 10\}$, $\{9, 12, 15\}$, $\{12, 16, 20\}$, $\{15, 20, 25\}$.
• Family $\{5, 12, 13\}$: Multiples include $\{10, 24, 26\}$.
• Family $\{8, 15, 17\}$: Multiples include $\{16, 30, 34\}$.
• Family $\{7, 24, 25\}$: Primitive set.

If legs are 9 and 12, recognize the ratio $3 : 4$, indicating the hypotenuse must be $5 \times 3 = 15$, avoiding squaring and root operations.

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Data Interpretation: Reading Tables, Charts, and Graphs

Data interpretation items evaluate your ability to synthesize information presented in visual and tabular formats.

Formats and Scanning Rules

1. Data Tables: Always verify row and column headers and note measurement units (e.g., "in thousands", "in millions of PHP").
2. Bar Charts: Used for discrete categorical comparisons. Check the baseline: if the vertical axis begins at a non-zero value, bar heights exaggerate relative differences.
3. Line Graphs: Used for continuous time-series data. The slope between two points indicates the rate of change; steeper lines represent faster growth or decline.
4. Pie Charts (Circle Graphs): Sectors represent parts of a whole ($100\%$ or $360^\circ$).
   • Conversion factor: $1\% = 3.6^\circ$, and $10\% = 36^\circ$.
   • To convert degrees ($\theta$) into percentage: $P = \frac{\theta}{360^\circ} \times 100\% = \frac{\theta}{3.6^\circ}$.
   • To convert percentage into degrees: $\theta = P \times 3.6^\circ$.

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Descriptive Statistics: Measures of Central Tendency and Dispersion

1. Arithmetic Mean (Average)

$\bar{x} = \frac{\sum x}{n} = \frac{x_1 + x_2 + \dots + x_n}{n}$

2. Median

The middle value when data points are arranged in ascending order:

• If $n$ is odd, the median is the single middle element at position $\frac{n + 1}{2}$.
• If $n$ is even, the median is the arithmetic mean of the two middle elements at positions $\frac{n}{2}$ and $\frac{n}{2} + 1$.
• The median is resistant to extreme outliers, making it superior to the mean when distributions are heavily skewed (such as household income statistics).

3. Mode and Range

• Mode: The value that appears with the greatest frequency. A distribution may have one mode (unimodal), two modes (bimodal), multiple modes, or no mode if all values appear with equal frequency.
• Range: The simplest measure of dispersion: $\text{Range} = x_{\text{max}} - x_{\text{min}}$.

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Step-by-Step Worked Problems

Worked Example 1: Composite Area and the Pythagorean Theorem

Problem: A school community garden in Laguna is designed in the shape of a right-angled triangle. Its two perpendicular legs measure 12 meters and 16 meters. A circular concrete composting station with a radius of 2 meters is constructed inside the garden. What is the length of the perimeter boundary fence along the hypotenuse, and what is the remaining plantable soil area of the garden? (Use $\pi \approx 3.14$).

Step-by-Step Solution:

1. Determine the hypotenuse using Pythagorean triples:
   • The legs are 12 and 16. Divide both by their greatest common factor, 4: $\frac{12}{4} = 3$ and $\frac{16}{4} = 4$.
   • This is a scaled $\{3, 4, 5\}$ triple with scale factor $k = 4$.
   • Hypotenuse $= 5 \times 4 = 20\text{ meters}$.
2. Calculate the total area of the triangular garden: $A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times 16 = 6 \times 16 = 96\text{ m}^2$
3. Calculate the area occupied by the circular composting station: $A_{\text{circle}} = \pi r^2 = 3.14 \times 2^2 = 3.14 \times 4 = 12.56\text{ m}^2$
4. Calculate the remaining plantable soil area: $A_{\text{plantable}} = A_{\text{triangle}} - A_{\text{circle}} = 96 - 12.56 = 83.44\text{ m}^2$

Worked Example 2: Target Score Using the Deviation Method

Problem: A student completes four periodic assessment quizzes in Numerical Reasoning, scoring 82, 87, 89, and 84. What score must the student achieve on the fifth and final quiz to obtain an overall average score of exactly 86 across all five assessments?

Step-by-Step Solution:

1. Method A — Total Points Architecture:
   • Target total points required: $5 \times 86 = 430\text{ points}$.
   • Current total points earned: $82 + 87 + 89 + 84 = 342\text{ points}$.
   • Required fifth score: $430 - 342 = 88\text{ points}$.
2. Method B — Deviation Architecture (Mental Shortcut):
   • Calculate deviations of each known quiz from target 86:
     • Quiz 1: $82 - 86 = -4$
     • Quiz 2: $87 - 86 = +1$
     • Quiz 3: $89 - 86 = +3$
     • Quiz 4: $84 - 86 = -2$
   • Sum the deviations: $(-4) + (+1) + (+3) + (-2) = -2$.
   • To achieve an overall net deviation of 0, Quiz 5 must provide a balance of $+2$.
   • Required fifth score: $86 + 2 = 88\text{ points}$.