IB MYP 4-5 Maths- Enlargement by a rational factor- Study Notes - New Syllabus
IB MYP 4-5 Maths- Enlargement by a rational factor – Study Notes
Extended
- Enlargement by a rational factor
IB MYP 4-5 Maths- Enlargement by a rational factor – Study Notes – All topics
Enlargement by a Rational Factor
Enlargement by a Rational Factor
Enlargement by a rational factor means the scale factor \(k\) is a fraction (e.g., \( \frac{1}{2}, \frac{3}{4} \)), which changes the size of a figure proportionally around a center of enlargement.
Key Points:
- If \( 0 < k < 1 \), the image is smaller than the original (reduction).
- If \( k > 1 \), the image is larger (enlargement).
- The method is the same: multiply distances from the center by \(k\).
- All angles remain unchanged, and the shape stays similar.
Steps:
- Identify the center of enlargement and the scale factor \(k\).
- Measure the distance of each vertex from the center.
- Multiply each distance by \(k\).
- Mark the new points and join them to form the image.
Example:
A triangle PQR is enlarged about point Z with coordinates (1, 2) by a scale factor of \( \dfrac{1}{2} \). The coordinates of Q are (7, 4). Find the new coordinates of Q after enlargement.
▶️ Answer/Explanation
Step 1: Center of enlargement \( Z(1, 2) \), Q is at \( (7, 4) \), and \( k = \dfrac{1}{2} \).
Step 2: Find displacement of Q from Z:
\( \Delta x = 7 – 1 = 6 \), \( \Delta y = 4 – 2 = 2 \).
Step 3: Multiply by \( k = \dfrac{1}{2} \):
\( \Delta x’ = 6 \times \dfrac{1}{2} = 3 \), \( \Delta y’ = 2 \times \dfrac{1}{2} = 1 \).
Step 4: Add back to center:
New Q’ = \( (1+3, 2+1) = (4, 3) \).
Answer: Q’ is at \( (4, 3) \).
Example:
A rectangle with vertices at (2, 3), (6, 3), (6, 7), and (2, 7) is enlarged by a scale factor of \( \dfrac{3}{4} \) about the origin (0, 0). Find the new coordinates.
▶️ Answer/Explanation
Step 1: Scale factor \( k = \dfrac{3}{4} \).
Step 2: Multiply each coordinate by \( k \):
\((2, 3) \rightarrow (2 \times \dfrac{3}{4}, 3 \times \dfrac{3}{4}) = (1.5, 2.25)\)
\((6, 3) \rightarrow (6 \times \dfrac{3}{4}, 3 \times \dfrac{3}{4}) = (4.5, 2.25)\)
\((6, 7) \rightarrow (6 \times \dfrac{3}{4}, 7 \times \dfrac{3}{4}) = (4.5, 5.25)\)
\((2, 7) \rightarrow (2 \times \dfrac{3}{4}, 7 \times \dfrac{3}{4}) = (1.5, 5.25)\)
Answer: New rectangle vertices are (1.5, 2.25), (4.5, 2.25), (4.5, 5.25), (1.5, 5.25).
Example:
A triangle ABC is enlarged by a scale factor of \( \dfrac{1}{2} \) about point O (the origin). The coordinates of A, B, and C are A(4, 2), B(6, 6), and C(8, 2). Find the new coordinates after enlargement and show the diagram.
▶️ Answer/Explanation
Step 1: Scale factor \( k = \dfrac{1}{2} \).
Step 2: Multiply each coordinate by \( \dfrac{1}{2} \):
A(4, 2) → A'(4 × \( \dfrac{1}{2} \), 2 × \( \dfrac{1}{2} \)) = (2, 1)
B(6, 6) → B'(6 × \( \dfrac{1}{2} \), 6 × \( \dfrac{1}{2} \)) = (3, 3)
C(8, 2) → C'(8 × \( \dfrac{1}{2} \), 2 × \( \dfrac{1}{2} \)) = (4, 1)
Step 3: Plot these new points and connect them.
Answer: The new triangle A’B’C’ has vertices at (2, 1), (3, 3), and (4, 1).
Important Notes:
- Fractional scale factors produce a smaller image.
- Apply the same method for any rational number.