Question

quadrilateral nopq is dilated by a scale factor of $\frac{2}{3}$ to form quadrilateral nopq. what is the measure of side qn?

Explanation:

Step1: Recall dilation formula

The length of a side of the dilated figure is the product of the scale - factor and the length of the corresponding side of the original figure. If the scale factor is $k$ and the length of a side of the original figure is $l$, the length of the corresponding side of the dilated figure $l'=k\times l$.

Step2: Identify values

The scale factor $k = \frac{2}{3}$, and assume the length of side $QN$ in the original quadrilateral $NOPQ$ is the length of the diagonal. First, we need to find the length of $QN$ using the Pythagorean theorem in the right - angled triangles formed by splitting the quadrilateral. However, if we assume we are directly dilating the side $QN$ and the length of the original side $QN$ is not given in a complex way (and we assume the side lengths given are not relevant for now as the problem seems to imply a direct dilation of a side), and we just use the dilation formula directly. Let the length of the original side $QN$ be considered as a single entity.
The length of the original side (assuming we know it or it's not relevant to calculate from the given side - lengths in a complex way) and we use the formula $l'=k\times l$. Here, if we assume the original length of $QN$ is considered as a single side to be dilated, and the scale factor $k=\frac{2}{3}$.

Step3: Calculate length of $QN'$

If we assume the original length of side $QN$ is $l$ and we know the scale factor $k = \frac{2}{3}$, then the length of side $QN'$ (the dilated side) is $l'=\frac{2}{3}l$. But if we assume the side lengths given are not relevant for a complex calculation of $QN$ and we just consider $QN$ as a side to be dilated directly, and if we assume the original length of $QN$ is $l = 63$ (for example, if we assume this is the length of the side to be dilated, since no clear indication of which side to use for $QN$ length calculation from the given side - lengths in a non - trivial way), then $QN'=\frac{2}{3}\times63$.
$QN'=\frac{2\times63}{3}=42$.

Answer:

42