Question: create a precise shape graph measurement for this scaling of simple shapes that are described by vector co-ordinates. Task 3.4: Scaling Shapes Using Vector Coordinates
Original Coordinates
Point Coordinate
A (1,1)
B (3,1)
C (4,3)
D (2,5)
E (0,3)
Scale Factor:
k=1.5
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Step 1: Use the Scaling Formula
$(x,y)\to(1.5x,1.5y)$
A
$(1,1)\to(1.5,1.5)$
B
$(3,1)\to(4.5,1.5)$
C
$(4,3)\to(6,4.5)$
D
$(2,5)\to(3,7.5)$
E
$(0,3)\to(0,4.5)$
New Coordinates
Point Scaled Coordinate
A' (1.5,1.5)
B' (4.5,1.5)
C' (6,4.5)
D' (3,7.5)
E' (0,4.5)
the Application in Software Interfaces
There are variety of applications for scaling, including:
Zooming maps
Resizing icons
User interface design
Mobile applications
Dashboard graphics (Foley et al., 1996).
If the scale factor is larger than 1, the object will be larger but will still have the same proportions. If the scale factor is below 1 the object is smaller. This keeps the graphics in line with one another on various screen sizes and resolutions (Foley et al., 1996).
Conclusion
The scaled coordinates are:
A'(1.5,1.5),B'(4.5,1.5),C'(6,4.5),D'(3,7.5),E'(0,4.5)
The need to scale is crucial for graphical rendering and user interface development, changing the size while maintaining the proportions of the shape (Foley et al., 1996).
1. **State the problem:** We are given original coordinates of points $A(1,1)$, $B(3,1)$, $C(4,3)$, $D(2,5)$, and $E(0,3)$ and a scale factor $k=1.5$. We need to find the new coordinates after scaling the shape by this factor.
2. **Scaling formula:** The scaling of a point $(x,y)$ by a factor $k$ is given by:
$$ (x,y) \to (kx, ky) $$
This means each coordinate is multiplied by the scale factor.
3. **Apply the formula to each point:**
- For $A(1,1)$:
$$ (1,1) \to (1.5 \times 1, 1.5 \times 1) = (1.5, 1.5) $$
- For $B(3,1)$:
$$ (3,1) \to (1.5 \times 3, 1.5 \times 1) = (4.5, 1.5) $$
- For $C(4,3)$:
$$ (4,3) \to (1.5 \times 4, 1.5 \times 3) = (6, 4.5) $$
- For $D(2,5)$:
$$ (2,5) \to (1.5 \times 2, 1.5 \times 5) = (3, 7.5) $$
- For $E(0,3)$:
$$ (0,3) \to (1.5 \times 0, 1.5 \times 3) = (0, 4.5) $$
4. **Summary of new coordinates:**
$$ A'(1.5, 1.5), B'(4.5, 1.5), C'(6, 4.5), D'(3, 7.5), E'(0, 4.5) $$
5. **Explanation:** Scaling by a factor greater than 1 enlarges the shape while preserving its proportions. Each point moves away from the origin proportionally.
6. **Applications:** This scaling method is used in zooming maps, resizing icons, user interface design, mobile apps, and dashboard graphics to maintain consistent proportions across different screen sizes.
**Final answer:** The scaled coordinates are $A'(1.5,1.5)$, $B'(4.5,1.5)$, $C'(6,4.5)$, $D'(3,7.5)$, and $E'(0,4.5)$.