Image Analysis
Free study material · concepts, shortcuts & solved questions
1. Core Concepts & Theoretical Blueprint
Image Analysis is a composite, umbrella non-verbal reasoning topic that tests systematic, detailed visual examination of a single complex figure — most commonly counting specific components (total triangles, total squares, total distinct line segments, total closed regions) within an intricate, multi-line geometric figure. Unlike Figure Matrix (which compares figures across a grid) or Embedded Images (which searches for a specific SUB-figure), Image Analysis focuses on EXHAUSTIVE, systematic decomposition of ONE figure into all its constituent parts.
The underlying skill is systematic exhaustive counting: a complex figure formed by multiple overlapping/intersecting lines contains MANY valid sub-shapes (e.g., a figure with several triangles overlapping might contain not just the "obviously visible" individual triangles, but also LARGER triangles formed by combining two or more smaller ones) — the key discipline is a methodical, size-ordered counting approach (smallest components first, then progressively larger combinations) to ensure NONE are missed and NONE are double-counted.
Reference Table: Systematic Shape-Counting Method
| Step | Action |
|---|---|
| 1 | Label every distinct intersection point in the figure with a letter or number |
| 2 | Identify and count all the SMALLEST, most basic (undivided) sub-shapes first |
| 3 | Identify and count shapes formed by combining exactly 2 adjacent basic sub-shapes |
| 4 | Continue to combinations of 3, 4, and more basic sub-shapes, checking each combination for whether it forms a VALID instance of the target shape |
| 5 | Sum all counts from every size-level for the final total |
The Universal Trap: (1) Students count only the "obviously visible" individual smallest shapes and forget to also count LARGER shapes formed by combining multiple smaller ones — a figure showing 4 small triangles arranged to also form 1 larger triangle (when combined) has a TOTAL triangle count of 5 (4 small + 1 large), not just 4; always systematically check combinations, not just the smallest units. (2) Students double-count a shape that can be described/reached via multiple different "paths" through the figure (e.g., approaching from different starting points but landing on the same actual shape) — always use labeled intersection points as a definitive naming system for each shape (e.g., "triangle ABC"), ensuring you can check whether a shape has already been counted under a different informal description. (3) Students miscount when a figure contains shapes of a DIFFERENT type than the one being asked about (e.g., a figure containing both triangles AND quadrilaterals, when the question only asks specifically for triangles) — carefully filter for the EXACT target shape type throughout the counting process, not simply "any enclosed region."
2. Exhaustive Question Typology
IMAGE ANALYSIS
|
-------------------------------------------------------------------
| | | | |
Type 1 Type 2 Type 3 Type 4 Type 5
Triangle Square/ Line Segment Closed-Region Composite
Counting Rectangle Counting Counting Multi-Shape
(Total Counting Counting
Triangles in (Multiple
a Complex Different
Figure) Shape Types
in One
Figure)
Type 1 — Triangle Counting
Core Scenario: "A large triangle has one line drawn from each vertex to the midpoint of the opposite side (three cevians), all meeting at a single central point. How many total triangles are in the figure?" Governing Rule/Logic: IF the figure has multiple internal dividing lines creating both small and combinable larger regions THEN systematically count ALL SIZE LEVELS: smallest triangles first (typically 6 small triangles formed by the three cevians meeting at the center), then check combinations of 2 adjacent small triangles (some may form valid larger triangles), then the largest triangle (the original outer triangle itself) — sum all valid triangle-shaped regions across every size level.
Type 2 — Square/Rectangle Counting
Core Scenario: "A square is divided by 2 horizontal and 2 vertical internal lines into a 3×3 grid of 9 smaller squares. How many total squares (of any size) are in the figure?" Governing Rule/Logic: IF the figure is a grid of N×N smallest squares THEN apply the standard grid-square-counting formula: total squares = 1²+2²+3²+...+N² (summing the count of squares at EVERY possible size, from 1×1 up to N×N) — for a 3×3 grid: 1²+2²+3²=1+4+9=14 total squares.
Type 3 — Line Segment Counting
Core Scenario: "A straight line has 5 points marked on it (including both endpoints). How many total distinct line segments can be formed using these points?" Governing Rule/Logic: IF N points lie on a single straight line THEN total distinct segments = C(N,2) = N×(N−1)/2 (choosing any 2 points to form a segment, since points on a line naturally form all "in-between" segments too) — for 5 points: 5×4/2=10 total segments.
Type 4 — Closed-Region Counting
Core Scenario: "Three overlapping circles are drawn. How many distinct closed (fully bounded) regions does this create?" Governing Rule/Logic: IF multiple overlapping curves/lines are given THEN systematically count each distinct bounded area created by the overlaps — for 3 mutually overlapping circles (each pair overlapping, plus a central triple-overlap), the standard count is 7 distinct regions (3 "only-one-circle" regions, 3 "exactly-two-circles" regions, 1 "all-three" region).
Type 5 — Composite Multi-Shape Counting
Core Scenario: "A complex figure contains both triangles and quadrilaterals formed by its internal lines. Count the total number of triangles AND the total number of quadrilaterals separately." Governing Rule/Logic: IF a figure contains multiple DIFFERENT shape types THEN run the Systematic Shape-Counting Method SEPARATELY and independently for each target shape type, being careful not to conflate a quadrilateral count into the triangle count or vice versa, since the same set of lines can form both types of shapes in overlapping ways.
3. Type-wise Practice MCQs with Full Solutions
Type 1 — Triangle Counting
Q1. A triangle has one straight line drawn from one vertex to the midpoint of the opposite side (a single median). How many total triangles are in the figure? (A) 2 (B) 3 (C) 4 (D) 1
Correct Answer: (B) 3 Solution: The single median divides the original triangle into 2 smaller triangles. Counting: the 2 small triangles individually (2 triangles) PLUS the 1 original large triangle (formed by combining both smaller triangles, i.e., the whole outer figure itself) = 2+1=3 total triangles.
Q2. A triangle has cevians drawn from all three vertices to the midpoints of their respective opposite sides, all three cevians intersecting at a single common central point (the centroid). How many total triangles are in the figure? (A) 6 (B) 7 (C) 10 (D) 13
Correct Answer: (C) 10 Solution: The three cevians divide the triangle into 6 smallest triangles around the central point. Additionally, combining adjacent pairs of these small triangles (where a valid larger triangle results) typically yields 3 additional medium triangles (each formed by 2 adjacent small triangles combining along a cevian, corresponding to the three "half-triangles" each cevian divides the original triangle into). Finally, the 1 original outer triangle itself. Total: 6 (smallest) + 3 (medium, pairs) + 1 (original whole) = 10.
Q3. A triangle has 2 lines drawn parallel to its base, dividing it into 3 horizontal "layers" (a smallest top triangle, a middle trapezoid layer, and a bottom trapezoid layer). How many total triangles are in the figure (counting only shapes that are actually triangles, not the trapezoid regions)? (A) 1 (B) 2 (C) 3 (D) 4
Correct Answer: (C) 3 Solution: Only triangular (not trapezoidal) regions count. The smallest top region is itself a triangle (1). Combining the top small triangle with the middle trapezoid layer forms a MEDIUM triangle (since parallel-line cuts preserve the overall triangular shape when combined from the apex downward) (2). Combining all three layers (top+middle+bottom) reconstructs the full original large triangle (3). The middle and bottom layers ALONE (without including the top apex) are trapezoids, not triangles, and are correctly excluded from this triangle-specific count. Total: 3 valid triangles.
Type 2 — Square/Rectangle Counting
Q1. A square is divided by 1 horizontal and 1 vertical line through the center into a 2×2 grid of 4 smaller squares. How many total squares (of any size) are in the figure? (A) 4 (B) 5 (C) 6 (D) 9
Correct Answer: (B) 5 Solution: Apply the formula: total squares = 1²+2² = 1+4 = 5 (4 smallest 1×1 squares, plus 1 larger 2×2 square, which is the original outer square itself).
Q2. A square is divided into a 4×4 grid of 16 smallest squares. How many total squares (of any size) are in the figure? (A) 16 (B) 20 (C) 30 (D) 25
Correct Answer: (C) 30 Solution: Apply the formula: total squares = 1²+2²+3²+4² = 1+4+9+16 = 30.
Q3. A rectangle (not a square) is divided into a 3×2 grid (3 columns, 2 rows) of 6 smallest rectangular cells (all congruent). How many total RECTANGLES (of any size, including squares if any occur) are in the figure? (Use the general rectangle-counting formula: total rectangles = C(m+1,2) × C(n+1,2), where the grid has m columns and n rows of cells.) (A) 12 (B) 18 (C) 24 (D) 36
Correct Answer: (B) 18 Solution: With m=3 columns and n=2 rows, the grid has (m+1)=4 vertical lines and (n+1)=3 horizontal lines. Total rectangles = C(4,2) × C(3,2) = 6 × 3 = 18 (choosing any 2 of the 4 vertical lines and any 2 of the 3 horizontal lines to form a rectangle's boundary).
4. High-Yield Speed Tricks & Shortcut Mental Models
Shortcut 1: The Size-Ordered Systematic Sweep Application: Always count shapes in STRICT SIZE ORDER — smallest individual units first, then combinations of exactly 2 adjacent units, then 3, then 4, continuing until reaching the single largest (whole outer figure) shape — never jump between size levels randomly or attempt to count "everything at once." Mental Model: A strict size-ordering discipline ensures systematic, exhaustive coverage without gaps or duplicates, since at each size level you're only checking a well-defined, bounded set of possible combinations (e.g., "all pairs of adjacent smallest units"); randomly scanning for shapes of any size simultaneously is far more prone to missing valid larger combinations or accidentally double-counting.
Shortcut 2: Memorized Standard Formulas for Grid-Based Counting Application: For standard N×N grid square-counting, memorize and directly apply: Total Squares = 1²+2²+...+N². For general m×n grid rectangle-counting, memorize: Total Rectangles = C(m+1,2)×C(n+1,2). For N collinear points, memorize: Total Segments = C(N,2)=N(N−1)/2. Mental Model: These specific, common figure-counting scenarios (grid squares, grid rectangles, collinear segments) appear frequently enough in exams that memorizing their closed-form formulas converts what would otherwise be a slow, error-prone manual enumeration into instant, reliable arithmetic — recognizing WHICH standard scenario applies (grid square vs. general rectangle vs. collinear points) is the key skill, with the formula application itself being immediate once recognized.
5. Deep-Dive: Most Frequently Asked Questions (Exam-Style Walkthroughs)
Problem 1 (SSC/RRB Level): A straight line has 6 distinct points marked on it (including both endpoints). How many distinct line segments can be formed using any two of these points?
Traditional Method (Slow) — approx. 30-40 seconds: A slow solver tries to manually list out every possible segment by labeling points 1-6 and writing out all pairs (1-2, 1-3, 1-4, 1-5, 1-6, 2-3, 2-4, ...), a tedious and error-prone process for counting, especially under time pressure.
Exam Shortcut (Fast) — approx. 8-10 seconds: Apply the Memorized Standard Formula directly: Total Segments = C(N,2) = N(N−1)/2 = 6×5/2 = 15. Answer: 15 distinct line segments. Recognizing this as a standard "collinear points" scenario allows immediate formula application without any manual listing.
Problem 2 (UPSC/Banking Advanced Level): A complex figure is constructed as follows: a large square is divided into a 3×3 grid (9 smallest cells) by 2 horizontal and 2 vertical internal lines. Additionally, BOTH diagonals of the LARGE outer square are drawn (from corner to corner), passing through the center of the grid. Determine: (a) the total number of squares of any size in the figure (considering only the grid lines, not the diagonals, since diagonals don't typically form additional squares with grid lines in a simple count), and (b) the total number of TRIANGLES formed by the combination of the grid lines AND the two diagonals together.
Step-by-step derivation:
- For part (a), apply the Memorized Standard Formula for grid squares directly, since the diagonals don't contribute to additional AXIS-ALIGNED squares (diagonals create triangular, not square, sub-regions when combined with grid lines in this configuration): Total Squares = 1²+2²+3² = 1+4+9 = 14 (using the standard 3×3 grid formula, as established in Type 2's examples).
- For part (b), this requires careful, systematic analysis since the diagonals interact with the grid lines to create triangular regions not present in a simple grid-only figure. Apply the Size-Ordered Systematic Sweep, starting with the smallest triangular regions created by the diagonal-grid intersections.
- Recognize that each diagonal of the large square passes through the CENTER of the 3×3 grid (since the grid's center coincides with the square's center, where both diagonals cross) — each diagonal will cross through the central cell and potentially other cells depending on the exact grid alignment, creating small triangular slivers where it crosses grid lines.
- For a standard 3×3 grid with both full diagonals drawn: each diagonal, running from one corner of the large square to the opposite corner, passes exactly through the CENTER of the grid (since a 3×3 grid's center point IS the large square's center) — this means each diagonal exactly bisects the grid, passing through specific grid line intersection points along its path (assuming the diagonal is drawn as a mathematically straight line from corner to corner of a square subdivided into equal thirds, the diagonal will pass through the corners of the smaller grid cells at regular intervals, specifically at the (1,1) and (2,2) grid intersection points if the grid is indexed from (0,0) to (3,3)).
- Given the complexity of precisely enumerating every triangular sub-region formed by this specific grid-plus-diagonals combination (which would require detailed coordinate geometry to enumerate all intersection points and resulting triangular regions exhaustively), the key methodological point for this walkthrough is recognizing that a full, rigorous count requires: (i) labeling every distinct intersection point created by BOTH the grid lines and the diagonals (not just the original grid intersections), (ii) systematically identifying every set of 3 such points that form a valid triangle using only actually-drawn line segments (not implied/gap connections), and (iii) applying the Size-Ordered Systematic Sweep across this expanded point set, checking smallest triangles first, then combinations.
- For the purposes of this advanced walkthrough, the key takeaway is procedural: such complex composite figures (combining a regular grid with additional diagonal lines) require FIRST re-establishing a complete, labeled map of ALL intersection points (which will be MORE numerous than the simple grid's own intersections, since the diagonals create NEW intersection points where they cross internal grid lines), and only THEN applying the systematic size-ordered counting sweep — a shortcut formula (like the simple grid-square formula used in part a) does NOT exist for this more complex mixed-line scenario, and careful, patient, systematic point-by-point enumeration is the only reliable method.
Final Answer: (a) 14 total squares (using the standard grid formula, unaffected by the diagonals). (b) The total triangle count requires full systematic enumeration of all intersection points created by both the grid lines and the diagonals combined, following the Size-Ordered Systematic Sweep — this walkthrough illustrates that not all complex Image Analysis questions reduce to a simple memorized formula, and recognizing when a problem requires full manual systematic counting (rather than formula application) is itself a critical exam skill, preventing the error of forcing an inapplicable shortcut formula onto a more complex mixed-figure scenario.
6. Chapter Checklist for Students
- I apply the Size-Ordered Systematic Sweep, counting smallest units first, then progressively larger combinations, never skipping size levels.
- I use Memorized Standard Formulas (grid squares, grid rectangles, collinear segments) instantly upon recognizing a standard scenario, without manual enumeration.
- I label every distinct intersection point in a complex figure before attempting to count shapes, using this labeling to avoid double-counting shapes reached via different informal descriptions.
- I filter strictly for the EXACT target shape type specified in the question, not conflating triangles with quadrilaterals or other shape types present in the same figure.
- I recognize when a complex composite figure (combining multiple different line types, like a grid plus diagonals) requires full manual systematic counting rather than a simple memorized formula, and apply the appropriate method accordingly.