Question

A rectangle has vertices E(4,8), F(2,8), G(2,-2) and H(-4,-2). The rectangle is dilated with the origin as the center of dilation so that G's is located at (5,5). Which algebraic reprensentation represents this dilation

Answer 1

Note: The image of G after the dilation must be G'(5,-5) instead of G'(5,5).

Given:

The vertices of a rectangle are E(4,8), F(2,8), G(2,-2) and H(-4,-2).

The rectangle is dilated with the origin as the center of dilation so that G's is located at (5,-5).

To find:

The algebraic representation that represents this dilation.

Solution:

If a figure is dilated by factor k with origin as the center of dilation, then the dilation is defined as:

$(x,y)\to (kx,ky)$              ...(i)

Let the given rectangle is dilated by factor k with origin as the center of dilation. Then,

$G(2,-2)\to G'(k(2),k(-2))$

$G(2,-2)\to G'(2k,-2k)$

The image of G after dilation is G'(5,-5). So,

$(2k,-2k)=(5,-5)$

On comparing both sides, we get

$2k=5$

$k=\dfrac{5}{2}$

So, the scale factor is $k=\dfrac{5}{2}$.

Substituting $k=\dfrac{5}{2}$ in (i), we get

$(x,y)\to \left(\dfrac{5}{2}x,\dfrac{5}{2}y\right)$

Therefore, the required algebraic representation to represents this dilation is $(x,y)\to \left(\dfrac{5}{2}x,\dfrac{5}{2}y\right)$.