Grade General Mathematics Term 3 Study Guide Flashcards

SECTION 1: CORE TRANSFORMATION RULES REFERENCE

Translation Rules
- A translation moves a point $(x, y)$ to a new position by adding or subtracting values from the coordinates: $(x \rightarrow x + h, y \rightarrow y + k)$ .
- Horizontal Movement ( $h$ ):
  - To move Right, add the number of units to the x-coordinate: $(x + h)$ .
  - To move Left, subtract the number of units from the x-coordinate: $(x - h)$ .
- Vertical Movement ( $k$ ):
  - To move Up, add the number of units to the y-coordinate: $(y + k)$ .
  - To move Down, subtract the number of units from the y-coordinate: $(y - k)$ .
Reflection Rules
- Across the x-axis: The x-coordinate remains the same, and the y-coordinate changes sign: $(x, -y)$ .
- Across the y-axis: The y-coordinate remains the same, and the x-coordinate changes sign: $(-x, y)$ .
Rotation Rules (About the Origin)
- $90^{\circ}$ Clockwise Rotation: Swap the coordinates and change the sign of the new y-coordinate: $(y, -x)$ .
- $180^{\circ}$ Clockwise Rotation: Keep the coordinates in the same order but change the signs of both: $(-x, -y)$ .
- $270^{\circ}$ Clockwise Rotation: Swap the coordinates and change the sign of the new x-coordinate: $(-y, x)$ .
Dilation Rules
- The transformation is defined by the rule $(kx, ky)$ , where $k$ is the scale factor.
- Enlargement: If the scale factor k > 1, the transformation results in an enlargement of the figure.
- Reduction: If the scale factor is between zero and one (0 < k < 1), the transformation results in a reduction of the figure.

SECTION 2: GEOMETRIC VOLUMES FORMULA SHEET

Volume of a Cylinder
- The volume is calculated as the area of the base ( $B$ ) multiplied by the height ( $h$ ).
- Formula: $V = B \times h$
- Expanded Formula: $V = \pi \times (r)^2 \times h$ , where $r$ is the base radius.
Volume of a Cone
- The volume of a cone is exactly one-third the volume of a cylinder with the same radius and height.
- Formula: $V = \frac{1}{3} \times \pi \times (r)^2 \times h$
Volume of a Sphere
- Formula: $V = \frac{4}{3} \times \pi \times (r)^3$
Volume of a Hemisphere
- The volume of a hemisphere is half the volume of a full sphere.
- Formula: $V = \frac{1}{2} \times (\frac{4}{3} \times \pi \times (r)^3) = \frac{2}{3} \times \pi \times (r)^3$

SECTION 3: CORE PRACTICE WORKED OUT EXAMPLES

Problem 1: Rotation Around a Vertex
- Scenario: Polygon $EFGH$ has vertices $E(-2, 1)$ , $F(-4, 3)$ , $G(-3, 3)$ , and $H(0, 0)$ . What are the coordinates after a $180^{\circ}$ rotation?
- Solution Procedure: Apply the $180^{\circ}$ rotation rule $(-x, -y)$ to each vertex.
  - Vertex $E(-2, 1) \rightarrow E'(2, -1)$
  - Vertex $F(-4, 3) \rightarrow F'(4, -3)$
  - Vertex $G(-3, 3) \rightarrow G'(3, -3)$
  - Vertex $H(0, 0) \rightarrow H'(0, 0)$
Problem 2: Composite Transformations
- Scenario: Consider rectangle $ABCD$ with coordinates $A(2, 8)$ , $B(10, 8)$ , $C(10, 0)$ , and $D(2, 0)$ . Translate the rectangle $2$ units right and $4$ units down, then rotate it $90^{\circ}$ clockwise about the origin.
- Step 1 (Translation): Apply the rule $(x + 2, y - 4)$ .
  - $A(2, 8) \rightarrow A'(2 + 2, 8 - 4) = (4, 4)$
  - $B(10, 8) \rightarrow B'(10 + 2, 8 - 4) = (12, 4)$
  - $C(10, 0) \rightarrow C'(10 + 2, 0 - 4) = (12, -4)$
  - $D(2, 0) \rightarrow D'(2 + 2, 0 - 4) = (4, -4)$
- Step 2 (Rotation): Apply the $90^{\circ}$ clockwise rule $(y, -x)$ to the translated points.
  - $A'(4, 4) \rightarrow A''(4, -4)$
  - $B'(12, 4) \rightarrow B''(4, -12)$
  - $C'(12, -4) \rightarrow C''(-4, -12)$
  - $D'(4, -4) \rightarrow D''(-4, -4)$
- Final Coordinates: $A''(4, -4)$ , $B''(4, -12)$ , $C''(-4, -12)$ , $D''(-4, -4)$ .
Problem 3: Indirect Measurement (Proportions)
- Scenario: A flag pole and a person stand next to each other. Set up a ratio to determine unknown heights based on shadow lengths: $\frac{\text{Height}}{\text{Shadow}} = \frac{\text{Height}}{\text{Shadow}}$ .
- Data Given:
  - Flag pole height: $21\,ft$
  - Flag pole shadow: $3\,ft$
  - Person shadow: $2.4\,ft$
- Calculation:
  - Ratio setup: $\frac{21}{3} = \frac{x}{2.4}$
  - Solve for $x$ : $7 = \frac{x}{2.4}$
  - $x = 7 \times 2.4 = 16.8\,ft$

SECTION 4: EXAM-STYLE PRACTICE MULTIPLE CHOICE BANK

(Q1) Which coordinate rotation describes a translation of 2 units right and 5 units down?
- A) $(x - 2, y + 5)$
- B) $(x + 2, y - 5)$
- C) $(x + 5, y - 2)$
- D) $(x - 5, y + 2)$
- Correct Answer: B
(Q2) If point A (2, -3) is reflected over the x-axis, what are the coordinates of A'?
- A) $(-2, -3)$
- B) $(2, 3)$
- C) $(3, 2)$
- D) $(-3, -2)$
- Correct Answer: B
(Q3) Triangle JKL has vertices J(5, 7), K(?, ?), and L(?, ?). Find Point J' after a $180^{\circ}$ clockwise rotation about the origin.
- A) $(-7, 5)$
- B) $(-5, -7)$
- C) $(5, -7)$
- D) $(-5, 7)$
- Correct Answer: B
(Q4) A figure is reflected over the y-axis. Find the coordinates of point B' if the preimage is B(-1, 2).
- A) $(-1, -2)$
- B) $(1, 2)$
- C) $(2, 1)$
- D) $(1, -2)$
- Correct Answer: B
(Q5) Calculate the scale factor (k) of a dilation from figure A to figure B if the preimage point is 4 and the image point is 1.6.
- A) $0.4$
- B) $2.5$
- C) $1.4$
- D) $0.2$
- Correct Answer: A
(Q6) Find the coordinate notation for a translation of 2 units left and 3 units up.
- A) $(x + 2, y + 3)$
- B) $(x - 2, y + 3)$
- C) $(x - 2, y - 3)$
- D) $(x + 3, y - 2)$
- Correct Answer: B
(Q7) Triangle ABC is congruent to Triangle DEF. Which statement must be true?
- A) Angle A corresponds to Angle E
- B) Angle C corresponds to Angle D
- C) Side AB matches Side DE
- D) Side BC matches Side DF
- Correct Answer: C
(Q8) Find the volume of a sphere with a radius of $6\,cm$ . (Use $\pi \approx 3.14$ )
- A) $150.72\,cm^3$
- B) $904.32\,cm^3$
- C) $113.04\,cm^3$
- D) $452.16\,cm^3$
- Correct Answer: B
(Q9) What range of scale factor K represents a reduction dilation?
- A) K > 1
- B) $K = 1$
- C) 0 < K < 1
- D) K < 0
- Correct Answer: C