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Counting Figures - Expert Level: rectangle counting EXPERT

Comprehensive self assessment worksheet covering 20 expert-level counting figures problems. Worksheet 28 of 30 emphasizes rectangle counting. Master image enumeration, design counting, pattern count through detailed explanations. Difficulty: challenging problems and time-bound practice. Tailored for expert-level preparation.

📝 Worksheet 28 of 30 • 20 questions • ⏱️ Estimated time: 20 minutes • 🎯 Expert level

What you'll learn in this worksheet:
Your progress through Counting Figures
Worksheet 28 of 30 (93% complete)

Question 1

Count the number of line segments in the following figure: +---+ |\ /| | X | |/ \| +---+
Square has 4 sides + 2 diagonals = 6. Plus the segments created by diagonal intersection? Actually, each diagonal is split into 2 segments, giving 4 more. Total = 8 line segments.
line, segment

Question 2

Count the number of line segments in the following figure: +---+ |\ /| | X | |/ \| +---+
Square has 4 sides + 2 diagonals = 6. Plus the segments created by diagonal intersection? Actually, each diagonal is split into 2 segments, giving 4 more. Total = 8 line segments.
line, segment

Question 3

Count the number of line segments in the following figure: o-o-o-o-o
With 5 points on a line, number of segments = C(5,2) = 10 segments.
line, segment

Question 4

Count the total number of triangles in the triangular grid below (1 row): /\ / \ / \ /______\
Upward pointing triangles: 1, Downward pointing triangles: 0. Total = 1 + 0 = 1
triangle, grid

Question 5

Count the number of triangles in the following figure: Triangular grid with 4 rows
In a 4-row triangular grid: Small triangles (16) + medium triangles of size 2 (7) + medium triangles of size 3 (3) + large triangle (1) = 27 total.
triangle, composite

Question 6

Count the total number of squares in the 4x4 grid below: +---+---+---+---+ | | | | | +---+---+---+---+ | | | | | +---+---+---+---+ | | | | | +---+---+---+---+ | | | | | +---+---+---+---+
In a 4x4 grid: 16 + 9 + 4 + 1 = 30
square, grid

Question 7

Count the number of triangles in the following figure: /\ /|\ / | \ /__|__\
The median divides the triangle into 2 smaller triangles. Total = 2 triangles.
triangle, basic

Question 8

Count the number of triangles in the following figure: /\ / \ / \ /______\
There is exactly 1 triangle in the figure.
triangle, basic

Question 9

Count the total number of rectangles (including squares) in the 3x3 grid below: +---+---+---+ | | | | +---+---+---+ | | | | +---+---+---+ | | | | +---+---+---+
Number of ways to choose 2 horizontal lines = C(4,2) = 6, Number of ways to choose 2 vertical lines = C(4,2) = 6. Total = 6 × 6 = 36
rectangle, grid

Question 10

Count the number of line segments in the following figure: o-o-o-o-o
With 5 points on a line, number of segments = C(5,2) = 10 segments.
line, segment

Question 11

Count the number of triangles in the following figure: Triangular grid with 4 rows
In a 4-row triangular grid: Small triangles (16) + medium triangles of size 2 (7) + medium triangles of size 3 (3) + large triangle (1) = 27 total.
triangle, composite

Question 12

Count the number of squares in the following figure: A 2x2 square grid with both diagonals in each small square: +---+---+ | | | +---+---+ | | | +---+---+
The 4 small squares (each 1x1) + 1 large square (2x2) = 5 squares. The diagonals do not create any additional squares, only triangles.
square, diagonals

Question 13

Count the number of triangles in the following figure: /\ /__\ /\ /\ /__\/__\
This creates 4 small congruent triangles. Total = 4 triangles.
triangle, basic

Question 14

Count the number of line segments in the following figure: +---+ |\ /| | X | |/ \| +---+
Square has 4 sides + 2 diagonals = 6. Plus the segments created by diagonal intersection? Actually, each diagonal is split into 2 segments, giving 4 more. Total = 8 line segments.
line, segment

Question 15

Count the total number of squares in the 2x2 grid below: +---+---+ | | | +---+---+ | | | +---+---+
In a 2x2 grid: 4 + 1 = 5
square, grid

Question 16

Count the number of triangles in the following figure: /\ /|\ / | \ / | \ / | \ /____|____\ \ | / \ | / \ | / \ | / \|/
The triangle has 3 cevians (lines from vertices to opposite sides). These create 6 smaller triangles within the large triangle. Total = 6 triangles.
triangle, composite

Question 17

Count the total number of parallelograms in the grid below (3 horizontal lines × 4 vertical lines): +---+---+---+ | | | | +---+---+---+ | | | | +---+---+---+
Choose 2 horizontal lines from 3: C(3,2) = 3, Choose 2 vertical lines from 4: C(4,2) = 6. Total = 3 × 6 = 18
parallelogram, grid

Question 18

Count the total number of parallelograms in the grid below (3 horizontal lines × 4 vertical lines): +---+---+---+ | | | | +---+---+---+ | | | | +---+---+---+
Choose 2 horizontal lines from 3: C(3,2) = 3, Choose 2 vertical lines from 4: C(4,2) = 6. Total = 3 × 6 = 18
parallelogram, grid

Question 19

Count the total number of squares in the 3x5 grid below: +---+---+---+---+---+ | | | | | | +---+---+---+---+---+ | | | | | | +---+---+---+---+---+ | | | | | | +---+---+---+---+---+
For a 3x5 grid: (3×5) + (2×4) + (1×3) = 26
square, rectangle

Question 20

Count the number of triangles in the following figure: /\ /|\ / | \ /__|__\ \ | / \ | / \|/
Two medians create 4 smaller triangles. Total = 4 triangles.
triangle, basic
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