IB MYP 4-5 Maths- Enlargement around a given point- Study Notes - New Syllabus
IB MYP 4-5 Maths- Enlargement around a given point – Study Notes
Extended
- Enlargement around a given point
IB MYP 4-5 Maths- Enlargement around a given point – Study Notes – All topics
Enlargement about a Point
Enlargement about a Point
Enlargement (or dilation) is a type of transformation that changes the size of a shape but keeps its shape and angles the same. The center of enlargement is a fixed point from which the figure is enlarged or reduced.
Key Features:
- The shape is scaled by a scale factor (k).
- All lengths are multiplied by k.
- If k > 1, the image is enlarged (bigger).
- If 0 < k < 1, the image is reduced (smaller).
- If k is negative, the image is on the opposite side of the center of enlargement.
- The center of enlargement is a fixed point from which distances are measured.
Properties of Enlargement:
| Property | Description |
|---|---|
| Shape | Remains similar (same angles, proportional sides) |
| Lengths | Multiplied by the scale factor k |
| Angles | Remain the same |
| Area | Multiplied by k² |
| Position | Depends on the center of enlargement and sign of k |
Steps for Enlargement about a Point
- Identify the center of enlargement.
-
- Decide the scale factor (k).
- For each vertex of the shape:
- Draw a straight line from the center to the vertex.
- Measure the distance from the center to the vertex and multiply by k.
- Mark the new point along the same line.
- Join the new points to get the enlarged image.
Example:
Triangle PQR is to be enlarged by a scale factor of \(\dfrac{1}{2}\), with point Z as the center of enlargement and with coordinates (1, 2).
What is the new coordinate of vertex Q in the enlargement?
▶️ Answer/Explanation
Step 1: Identify the displacement of Q from Z.
Vertex Q is 6 squares to the right and 2 squares above the center of enlargement (Z).
Step 2: Multiply this displacement by the scale factor:
\( \text{Horizontal: } 6 \times \dfrac{1}{2} = 3 \)
\( \text{Vertical: } 2 \times \dfrac{1}{2} = 1 \)
Step 3: From Z (1, 2), move 3 squares right and 1 square up:
\( \text{New coordinates of Q’} = (1+3, \; 2+1) = (4, 3) \)
Answer: Vertex Q’ has the coordinates (4, 3).
Example:
A triangle ABC is enlarged about point O with a scale factor of 2. If OA = 3 cm, OB = 4 cm, and OC = 5 cm, find the distances O’A’, O’B’, and O’C’.
▶️ Answer/Explanation
Step 1: Scale factor k = 2.
Step 2: Multiply each original distance by k:
OA’ = 3 × 2 = 6 cm
OB’ = 4 × 2 = 8 cm
OC’ = 5 × 2 = 10 cm
Answer: The new distances are 6 cm, 8 cm, and 10 cm.
Example: A square PQRS is enlarged about point O with a scale factor of -1.5. If OP = 4 cm, what is OP’?
▶️ Answer/Explanation
Step 1: Scale factor k = -1.5 (negative means opposite side of O).
Step 2: Multiply distance by |k|: OP’ = 4 × 1.5 = 6 cm.
Step 3: Place P’ on the opposite side of O along the same line.
Answer: OP’ = 6 cm on the opposite side of O.
Important Notes:
- If the center is inside the shape, the image expands or shrinks around that point.
- If the center is outside the shape, the image moves away from or towards the center along straight lines.
- Negative scale factor flips the image across the center.