Consider a triangle with co-ordinates points: A(0, 0), B(3, 3) and C(2, 5). When scaling parameter is 2 towards x-axis and 4 towards y-axis, then what are the new co-ordinates of triangle?

Question

Prepp Team · Accepted Answer

Understanding Triangle Scaling and New Coordinates

The problem asks us to find the new coordinates of a triangle after applying a scaling transformation. We are given the original coordinates of the vertices of the triangle: A(0, 0), B(3, 3), and C(2, 5). We are also given the scaling parameters: 2 towards the x-axis and 4 towards the y-axis. This means the scaling factor for the x-direction ($s_x$) is 2, and the scaling factor for the y-direction ($s_y$) is 4.

Geometric Scaling Transformation

Geometric scaling is a transformation that changes the size of an object. When scaling is performed relative to the origin (0,0), the new coordinates (x', y') of a point (x, y) are calculated using the following formula:

$(x', y') = (x \cdot s_x, y \cdot s_y)$

Where:

$x$ and $y$ are the original coordinates of the point.
$s_x$ is the scaling factor in the x-direction.
$s_y$ is the scaling factor in the y-direction.
$x'$ and $y'$ are the new coordinates after scaling.

In this specific problem, the scaling factors are $s_x = 2$ and $s_y = 4$.

Calculating New Coordinates for Each Vertex

We apply the scaling formula to each vertex of the triangle:

Scaling Vertex A(0, 0)

For point A(0, 0):

Original coordinates: $x = 0, y = 0$
Scaling factors: $s_x = 2, s_y = 4$
New coordinates: $A'(0 \cdot 2, 0 \cdot 4) = A'(0, 0)$

The new coordinates for vertex A are (0, 0).

Scaling Vertex B(3, 3)

For point B(3, 3):

Original coordinates: $x = 3, y = 3$
Scaling factors: $s_x = 2, s_y = 4$
New coordinates: $B'(3 \cdot 2, 3 \cdot 4) = B'(6, 12)$

The new coordinates for vertex B are (6, 12).

Scaling Vertex C(2, 5)

For point C(2, 5):

Original coordinates: $x = 2, y = 5$
Scaling factors: $s_x = 2, s_y = 4$
New coordinates: $C'(2 \cdot 2, 5 \cdot 4) = C'(4, 20)$

The new coordinates for vertex C are (4, 20).

New Coordinates of the Scaled Triangle

After applying the scaling transformation with $s_x = 2$ and $s_y = 4$ to each vertex, the new coordinates of the triangle are:

A': (0, 0)
B': (6, 12)
C': (4, 20)

Thus, the new coordinates of the triangle are A(0, 0), B(6, 12), and C(4, 20).

Original Vertex	Original Coordinates (x, y)	Scaling Factors ($s_x$, $s_y$)	New Coordinates ($x \cdot s_x$, $y \cdot s_y$)	New Vertex
A	(0, 0)	(2, 4)	($0 \cdot 2$, $0 \cdot 4$) = (0, 0)	A'
B	(3, 3)	(2, 4)	($3 \cdot 2$, $3 \cdot 4$) = (6, 12)	B'
C	(2, 5)	(2, 4)	($2 \cdot 2$, $5 \cdot 4$) = (4, 20)	C'

Revision Table: Triangle Scaling Basics

Concept	Description	Formula (relative to origin)
Scaling	Changing the size of an object.	$(x', y') = (x \cdot s_x, y \cdot s_y)$
Uniform Scaling	$s_x = s_y$ (Size changes, shape preserved)	$(x', y') = (x \cdot s, y \cdot s)$
Non-Uniform Scaling	$s_x \neq s_y$ (Size and shape may change)	$(x', y') = (x \cdot s_x, y \cdot s_y)$
Scaling Factors	Values $s_x, s_y$ determining the extent of scaling.	$s_x > 1$ for enlargement in x, $0 < s_x < 1$ for shrinking in x. Same for $s_y$.

Additional Information: Types of Transformations

Geometric transformations are operations that change the position, size, or shape of an object. Common 2D transformations include:

Translation: Moving an object to a new location without changing its size or orientation. $ (x', y') = (x + t_x, y + t_y) $
Rotation: Rotating an object about a point (usually the origin) by a certain angle.
Scaling: Changing the size of an object. Can be uniform or non-uniform.
Shearing: Skewing an object along an axis.

Scaling is a fundamental transformation in computer graphics and geometry.

Prepp Team · Answer

A(0, 0), B(3/2, 3/4), C(1, 5/4)

Prepp Team · Answer

A(0, 0), B(5, 7), C(4, 9)

Prepp Team · Answer

A(2, 4), B(5, 7), C(4, 9)