Comprehensive Study Notes: Geometry, Area of Basic and Composite Shapes, and Pythagorean Theorem Application

Television Screen Diagonal Calculation

• Problem Overview: Television screen sizes are defined by their diagonal measurement. For a screen with a width of $1.2\,\text{m}$ and a height of $0.8\,\text{m}$, the diagonal length is computed using the Pythagorean theorem.

• Geometric Principle: In any right-angled triangle formed by the width, height, and diagonal of a rectangle, the Pythagorean theorem states:   $a^2 + b^2 = c^2$   where $a$ represents the width, $b$ represents the height, and $c$ represents the diagonal length ($d$).

• Detailed Calculation Steps:

  • Define given measurements:     $\text{Width } (a) = 1.2\,\text{m}$$\text{Height } (b) = 0.8\,\text{m}$
  • Square both dimensions:     $a^2 = (1.2\,\text{m})^2 = 1.44\,\text{m}^2$$b^2 = (0.8\,\text{m})^2 = 0.64\,\text{m}^2$
  • Add the squared dimensions together:     $d^2 = 1.44 + 0.64 = 2.08\,\text{m}^2$
  • Take the square root of the sum:     $d = \sqrt{2.08} \approx 1.44222051\,\text{m}$

• Final Rounded Answers:

  • Rounded to the nearest whole metre:     $d \approx 1\,\text{m}$
  • Rounded to one decimal place:     $d \approx 1.4\,\text{m}$
  • Converted to centimetres ($1\,\text{m} = 100\,\text{cm}$):     $d \approx 144.22\,\text{cm}$

Area Calculations for Basic Geometric Shapes

• 2a. Parallelogram Area Calculation:

  • Properties: A parallelogram is a quadrilateral with two pairs of parallel opposite sides.
  • Given Values:$\text{Base } (b) = 6\,\text{mm}$$\text{Perpendicular Height } (h) = 7\,\text{mm}$
  • Formula:$\text{Area} = b \times h$
  • Computation:$\text{Area} = 6\,\text{mm} \times 7\,\text{mm} = 42\,\text{mm}^2$

• 2b. Rhombus / Diamond Area Calculation:

  • Properties: The area of a rhombus or kite is calculated using the lengths of its two perpendicular interior diagonals.
  • Given Values:$\text{Diagonal } 1\,(d_1) = 4\,\text{cm}$$\text{Diagonal } 2\,(d_2) = 6\,\text{cm}$
  • Formula:$\text{Area} = \frac{1}{2} \times d_1 \times d_2$
  • Computation:$\text{Area} = \frac{1}{2} \times 4\,\text{cm} \times 6\,\text{cm} = 12\,\text{cm}^2$

• 2c. Trapezium Area Calculation:

  • Properties: A trapezium (trapezoid) is a quadrilateral with at least one pair of parallel sides ($a$ and $b$).
  • Given Values:$\text{Top Parallel Side } (a) = 7\,\text{cm}$$\text{Bottom Parallel Side } (b) = 10\,\text{cm}$$\text{Perpendicular Height } (h) = 3\,\text{cm}$
  • Formula:$\text{Area} = \frac{1}{2} \times (a + b) \times h$
  • Computation:     \text{Area} = \frac{1}{2} \n\times (7\,\text{cm} + 10\,\text{cm}) \times 3\,\text{cm}     $\text{Area} = \frac{1}{2} \times 17\,\text{cm} \times 3\,\text{cm} = 25.5\,\text{cm}^2$

Area Calculations for Composite Shapes

• 3a. Symmetrical Cross Composite Area:

  • Structure Analysis: The cross shape is composed of $5$ identical, congruent squares arranged in a grid (one center square and four surrounding arm squares).
  • Determining Side Length:
  • The total span across three aligned square arms is $12\,\text{cm}$.
  • Side length of an individual square ($s$):       $s = \frac{12\,\text{cm}}{3} = 4\,\text{cm}$
  • Area Calculations:
  • Area of one individual square:       $\text{Area}_{\text{square}} = s^2 = (4\,\text{cm})^2 = 16\,\text{cm}^2$
  • Total area of all $5$ squares combined:       $\text{Area}_{\text{total}} = 5 \times 16\,\text{cm}^2 = 80\,\text{cm}^2$

• 3b. Right-Angled Trapezoid Composite Area:

  • Structure Analysis: The shape represents a right-angled trapezium with vertical parallel sides and a horizontal base.
  • Given Values:$\text{Left Vertical Height } (a) = 3.5\,\text{m}$$\text{Right Vertical Height } (b) = 2.0\,\text{m}$$\text{Horizontal Base Length } (h) = 7.0\,\text{m}$
  • Direct Trapezoid Calculation Method:$\text{Area} = \frac{1}{2} \times (a + b) \times h$$\text{Area} = \frac{1}{2} \times (3.5\,\text{m} + 2.0\,\text{m}) \times 7.0\,\text{m}$$\text{Area} = \frac{1}{2} \times 5.5\,\text{m} \times 7.0\,\text{m} = 19.25\,\text{m}^2$
  • Sub-Region Decomposition Method (Rectangle + Triangle):
  • Area of bottom rectangular section ($7.0\,\text{m} \times 2.0\,\text{m}$):       $\text{Area}_{\text{rectangle}} = 7.0\,\text{m} \times 2.0\,\text{m} = 14.0\,\text{m}^2$
  • Height of top triangle section:       $h_{\text{triangle}} = 3.5\,\text{m} - 2.0\,\text{m} = 1.5\,\text{m}$
  • Area of top right-triangle section:       $\text{Area}_{\text{triangle}} = \frac{1}{2} \times 7.0\,\text{m} \times 1.5\,\text{m} = 5.25\,\text{m}^2$
  • Combined Total Area:       $\text{Area}_{\text{total}} = 14.0\,\text{m}^2 + 5.25\,\text{m}^2 = 19.25\,\text{m}^2$

• 3c. Square with Inscribed Triangular Cutout Area:

  • Structure Analysis: An outer square contains an interior triangle sharing the bottom edge as its base.
  • Given Values:$\text{Square Side Length } (s) = 20\,\text{cm}$$\text{Triangle Base } (b) = 20\,\text{cm}$$\text{Triangle Height } (h) = 14\,\text{cm}$
  • Area Calculations:
  • Area of outer square:       $\text{Area}_{\text{square}} = s^2 = (20\,\text{cm})^2 = 400\,\text{cm}^2$
  • Area of inner triangle:       $\text{Area}_{\text{triangle}} = \frac{1}{2} \times b \times h = \frac{1}{2} \times 20\,\text{cm} \times 14\,\text{cm} = 140\,\text{cm}^2$
  • Area of remaining outer region (Square minus Triangle):       $\text{Area}_{\text{remaining}} = 400\,\text{cm}^2 - 140\,\text{cm}^2 = 260\,\text{cm}^2$

Area Analysis of Flag Design and Parallelogram Stripe

• 4a. Area of the Dark Grey Parallelogram Stripe:

  • Given Values:$\text{Stripe Base } (b) = 20\,\text{cm}$$\text{Flag / Stripe Height } (h) = 50\,\text{cm}$
  • Formula:$\text{Area} = b \times h$
  • Computation:$\text{Area}_{\text{stripe}} = 20\,\text{cm} \times 50\,\text{cm} = 1000\,\text{cm}^2$

• Full Geometric Analysis of Proposed Flag:

  • Total Width of Flag:$\text{Width} = 20\,\text{cm} + 70\,\text{cm} = 90\,\text{cm}$
  • Total Area of Flag:$\text{Area}_{\text{flag}} = 90\,\text{cm} \times 50\,\text{cm} = 4500\,\text{cm}^2$
  • Area of Remaining White Region:$\text{Area}_{\text{white}} = 4500\,\text{cm}^2 - 1000\,\text{cm}^2 = 3500\,\text{cm}^2$
  • Proportion of Flag Occupied by Dark Grey Stripe:$\text{Proportion} = \frac{1000\,\text{cm}^2}{4500\,\text{cm}^2} = \frac{2}{9} \approx 22.22\%$