# A rectangular room has coordinates Q(2, 2), R (7, 2), S(7, 5), and T(2, 5) on the blueprint. The homeowner wants this room to be 25% larger. What are the coordinates of the new room?

Jun 6, 2018

A few ambiguities in this problem. Clearly we're to dilate but it's not clear around which point. It's not clear if we're to scale the linear dimensions by $\frac{5}{4}$ -- maybe 25% larger refers to the area or the volume.

Disambiguating, let's say we'll keep Q the same and scale the linear dimensions, so we dilate by $\frac{5}{4}$ around Q.

We translate each point to the origin by subtracting $\left(2 , 2\right)$, then we scale by $\frac{5}{4}$, then we translate back:

$\left(2 , 2\right) \to \frac{5}{4} \left(\left(2 , 2\right) - \left(2 , 2\right)\right) + \left(2 , 2\right) = \left(2 , 2\right)$

$\left(7 , 5\right) \to \frac{5}{4} \left(\left(7 , 5\right) - \left(2 , 2\right)\right) + \left(2 , 2\right) = \left(\frac{33}{4} , \frac{23}{4}\right)$

The rest don't require calculation:

$\left(7 , 2\right) \to \left(\frac{33}{4} , 2\right)$

$\left(2 , 5\right) \to \left(2 , \frac{23}{4}\right)$