In triangle $PQR$, $U$ is a point on $PR$, $S$ is a point on $PQ$, $T$ is a point on $QR$ with $US$ being parallel to $RQ$, and $UT$ is parallel to $PQ$. The area of $PSU$ is $120 in^2$ and the area of triangle $TUR$ is $270 in^2$. The area of $QST$ in $in^2$, is
This is a multiple choice question
$A) 150$
$B) 160$
$C) 170$
$D) 180$
$E) 200$
I figure triangle $STU$ being equal to triangle $QST$ but $SU$ is not confirmed as equal to $QT$. Is there any way to confirm the lengths, maybe if I separate the areas of the two triangles to experiment with proportions since the two triangles with area have parallel sides equal to each other.
Triangles $UTR$ and $PSU$ are both similar to triangle $PQR$, and thus to each other. The ratio of their areas is $\frac{270}{120} = \frac94$, so the ratio of their bases $UR$ and $PU$ is $\sqrt{\frac94} = \frac32$. Since $PR = PU + UR$, the ratio of $PR$ to $PU$ is $\frac52$. Thus the area of triangle $PQR$ is $\frac{25}{4}$ times that of $PSU$: $\frac{25}{4}\cdot120 = 750$. Subtracting the two smaller triangles from the total, the area of parallelogram $SQTU$ is $750 - 120 - 270 = 360$. The area of triangle $QST$ is half that of the parallelogram, $\frac12\cdot360 = 180$, which is choice $D$.