Question

the points p(-9,4), q(0,-8), r(4,-5), and s(-5,7) form quadrilateral pqrs. plot the points then click the graph quadrilateral button. click on the graph to plot a point. click a point to delete it. use the dropdown menus and answer blanks below to prove the quadrilateral is a rectangle. answer attempt 1 out of 2 i will prove that quadrilateral pqrs is a rectangle by demonstrating that: adjacent sides are perpendicular both pairs of opposite sides are parallel one pair of opposite sides is of equal measure and parallel both pairs of opposite sides are of equal measure all sides are of equal measure and adjacent sides are perpendicular

Explanation:

Step1: Recall Rectangle Properties

A rectangle is a quadrilateral with both pairs of opposite sides parallel (so it's a parallelogram) and all angles are right angles (adjacent sides are perpendicular). Alternatively, a rectangle can be defined as a parallelogram with one right angle (so adjacent sides perpendicular) or a quadrilateral with both pairs of opposite sides parallel and adjacent sides perpendicular. The key property for proving a quadrilateral is a rectangle is that it is a parallelogram (both pairs of opposite sides parallel) and adjacent sides are perpendicular, or equivalently, both pairs of opposite sides are parallel and one pair of adjacent sides is perpendicular, or all sides have equal opposite sides and adjacent sides perpendicular. The most accurate way is to show both pairs of opposite sides are parallel (so it's a parallelogram) and adjacent sides are perpendicular (so angles are right angles). Looking at the options, the correct condition is "both pairs of opposite sides are parallel AND adjacent sides are perpendicular" (or the option that combines both: in the dropdown, the correct one is "both pairs of opposite sides are parallel AND adjacent sides are perpendicular" which is equivalent to "all sides are of equal measure AND adjacent sides are perpendicular" no, wait, no—wait, opposite sides of a rectangle are equal, adjacent sides are perpendicular. Wait, the options: let's re - examine. The options are:
- adjacent sides are perpendicular
- both pairs of opposite sides are parallel
- one pair of opposite sides is of equal measure AND parallel
- both pairs of opposite sides are of equal measure
- all sides are of equal measure AND adjacent sides are perpendicular

Wait, the correct property for a rectangle: a rectangle is a parallelogram (both pairs of opposite sides parallel) with adjacent sides perpendicular (so right angles). Also, in a rectangle, both pairs of opposite sides are equal in length. So the most accurate condition to prove a quadrilateral is a rectangle is that both pairs of opposite sides are parallel (so it's a parallelogram) and adjacent sides are perpendicular (so it has right angles). Among the given options, the one that fits is "both pairs of opposite sides are parallel AND adjacent sides are perpendicular" (which might be phrased as "all sides are of equal measure AND adjacent sides are perpendicular" no, that's a square. Wait, no: in a rectangle, opposite sides are equal, adjacent sides are perpendicular. So the correct option is the one that says both pairs of opposite sides are parallel (so parallelogram) and adjacent sides are perpendicular (so right angles). So when proving PQRS is a rectangle, we need to show that both pairs of opposite sides are parallel (so it's a parallelogram) and adjacent sides are perpendicular (so angles are right angles).

Step2: Analyze the Points

First, we can find the slopes of the sides. The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$.

For side $PQ$: $P(-9,4)$ and $Q(0,-8)$. Slope of $PQ$, $m_{PQ}=\frac{-8 - 4}{0 - (-9)}=\frac{-12}{9}=-\frac{4}{3}$.

For side $QR$: $Q(0,-8)$ and $R(4,-5)$. Slope of $QR$, $m_{QR}=\frac{-5 - (-8)}{4 - 0}=\frac{3}{4}$.

For side $RS$: $R(4,-5)$ and $S(-5,7)$. Slope of $RS$, $m_{RS}=\frac{7 - (-5)}{-5 - 4}=\frac{12}{-9}=-\frac{4}{3}$.

For side $SP$: $S(-5,7)$ and $P(-9,4)$. Slope of $SP$, $m_{SP}=\frac{4 - 7}{-9 - (-5)}=\frac{-3}{-4}=\frac{3}{4}$.

Now, we see that $m_{PQ}=m_{RS}=-\frac{4}{3}$, so $PQ\parallel RS$. And $m_{QR}=m_{SP}=\frac{3}{4}$, so $QR\parallel SP$. So both pairs of opposite sides are parallel (so PQRS is a parallelogram). Now, check the product of slopes of adjacent sides: $m_{PQ}\times m_{QR}=(-\frac{4}{3})\times\frac{3}{4}=- 1$. So $PQ\perp QR$. Since PQRS is a parallelogram and adjacent sides are perpendicular, it is a rectangle. So the condition to prove it's a rectangle is that both pairs of opposite sides are parallel (which we proved by equal slopes) and adjacent sides are perpendicular (which we proved by the product of slopes being - 1). So the correct option from the dropdown is the one that states both pairs of opposite sides are parallel and adjacent sides are perpendicular (or the equivalent phrasing in the options).

Step3: Conclusion

To prove quadrilateral $PQRS$ is a rectangle, we need to show that both pairs of opposite sides are parallel (which we did by calculating slopes: $PQ\parallel RS$ and $QR\parallel SP$) and adjacent sides are perpendicular (which we did by showing the product of slopes of adjacent sides is - 1, e.g., $m_{PQ}\times m_{QR}=-1$). So the correct condition from the dropdown is the one that combines both pairs of opposite sides being parallel and adjacent sides being perpendicular.

Answer:

The correct condition to prove quadrilateral \( PQRS \) is a rectangle is "both pairs of opposite sides are parallel AND adjacent sides are perpendicular" (or the equivalent option in the dropdown that represents this property, which can be verified by calculating the slopes of the sides as shown above: \( PQ\parallel RS \), \( QR\parallel SP \) (proving both pairs of opposite sides are parallel) and \( m_{PQ}\times m_{QR}=- 1 \), \( m_{QR}\times m_{RS}=-1 \), etc. (proving adjacent sides are perpendicular)).