Question

determining the name of a quadrilateral
what is the most specific name for quadrilateral
pqrs?
parallelogram
rectangle
rhombus
square

Explanation:

Step1: Calculate slopes of sides

To determine the type of quadrilateral, we first find the slopes of the sides. The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$.

- Slope of $PQ$: $P(4,9)$, $Q(-1,1)$
$m_{PQ} = \frac{1 - 9}{-1 - 4} = \frac{-8}{-5} = \frac{8}{5}$

- Slope of $QR$: $Q(-1,1)$, $R(-9,-4)$
$m_{QR} = \frac{-4 - 1}{-9 - (-1)} = \frac{-5}{-8} = \frac{5}{8}$

- Slope of $RS$: $R(-9,-4)$, $S(-4,4)$
$m_{RS} = \frac{4 - (-4)}{-4 - (-9)} = \frac{8}{5}$

- Slope of $SP$: $S(-4,4)$, $P(4,9)$
$m_{SP} = \frac{9 - 4}{4 - (-4)} = \frac{5}{8}$

Step2: Analyze parallel sides

From the slopes:
- $m_{PQ} = m_{RS} = \frac{8}{5}$, so $PQ \parallel RS$.
- $m_{QR} = m_{SP} = \frac{5}{8}$, so $QR \parallel SP$.

A quadrilateral with both pairs of opposite sides parallel is a parallelogram.

Step3: Check for rectangle, rhombus, square

- **Rectangle**: Requires adjacent sides to be perpendicular (slopes product = -1).
$m_{PQ} \times m_{QR} = \frac{8}{5} \times \frac{5}{8} = 1
eq -1$, so not a rectangle.

- **Rhombus**: Requires all sides to be equal in length.
Length of $PQ$: $\sqrt{(-1 - 4)^2 + (1 - 9)^2} = \sqrt{25 + 64} = \sqrt{89}$
Length of $QR$: $\sqrt{(-9 - (-1))^2 + (-4 - 1)^2} = \sqrt{64 + 25} = \sqrt{89}$
Wait, but wait—wait, let’s recalculate. Wait, no, actually, if we check slopes for perpendicularity (for rectangle) we saw they aren’t perpendicular, but wait, length of $PQ$ and $QR$: both $\sqrt{89}$? Wait, no, maybe I made a mistake. Wait, no—wait, the slope product was 1, meaning they are equal, so angles are not right angles. But wait, let's check side lengths again.

Wait, $PQ$: distance between (4,9) and (-1,1): $\sqrt{(-1 - 4)^2 + (1 - 9)^2} = \sqrt{(-5)^2 + (-8)^2} = \sqrt{25 + 64} = \sqrt{89}$

$QR$: distance between (-1,1) and (-9,-4): $\sqrt{(-9 - (-1))^2 + (-4 - 1)^2} = \sqrt{(-8)^2 + (-5)^2} = \sqrt{64 + 25} = \sqrt{89}$

$RS$: distance between (-9,-4) and (-4,4): $\sqrt{(-4 - (-9))^2 + (4 - (-4))^2} = \sqrt{5^2 + 8^2} = \sqrt{25 + 64} = \sqrt{89}$

$SP$: distance between (-4,4) and (4,9): $\sqrt{(4 - (-4))^2 + (9 - 4)^2} = \sqrt{8^2 + 5^2} = \sqrt{64 + 25} = \sqrt{89}$

Wait, all sides are equal? But then why isn't it a rhombus? Wait, no—wait, the slope of $PQ$ is 8/5, slope of $QR$ is 5/8. The product is 1, so they are equal, meaning the angles are 45-135? No, wait, if two lines have slopes $m$ and $n$, and $m \times n = 1$, they are congruent (equal angles with x-axis), so the angle between them is 0? No, wait, no—if slope1 is 8/5 and slope2 is 5/8, they are reciprocals, so the lines are symmetric with respect to the line $y=x$, so the angle between them is not 90 degrees. Wait, but if all sides are equal and opposite sides are parallel, it should be a rhombus. But wait, maybe I miscalculated the slopes. Wait, no—wait, let's re-express the points:

Wait, the quadrilateral is P(4,9), Q(-1,1), R(-9,-4), S(-4,4). Let's list the order: P to Q to R to S to P.

Wait, maybe the order is different? Wait, the graph shows the points connected as P-Q-R-S-P? Wait, no, the graph has P(4,9), Q(-1,1), R(-9,-4), S(-4,4), and back to P. So the sides are PQ, QR, RS, SP.

Wait, but when we calculated slopes, PQ and RS have slope 8/5, QR and SP have slope 5/8. So opposite sides are parallel (parallelogram). Now, check if adjacent sides are equal:

PQ: distance between (4,9) and (-1,1): $\sqrt{( -1 - 4)^2 + (1 - 9)^2} = \sqrt{25 + 64} = \sqrt{89}$

QR: distance between (-1,1) and (-9,-4): $\sqrt{( -9 +1)^2 + (-4 -1)^2} = \sqrt{64 +25} = \sqrt{89}$

RS: distance between (-9,-4) and (-4,4): $\sqrt{( -4 +9)^2 + (4 +4)^2} = \sqrt{25 +64} = \sqrt{89}$

SP: distance between (-4,4) and (4,9): $\sqrt{(4 +4)^2 + (9 -4)^2} = \sqrt{64 +25} = \sqrt{89}$

So all sides are equal. Then why isn't it a rhombus? Wait, maybe the angles are not 90 degrees, so it's a rhombus (since rhombus is a parallelogram with all sides equal). But the options include parallelogram, rectangle, rhombus, square. Wait, maybe I made a mistake in the slope product. Wait, slope of PQ is (1-9)/(-1-4) = (-8)/(-5) = 8/5. Slope of QR is (-4-1)/(-9 - (-1)) = (-5)/(-8) = 5/8. Product is (8/5)*(5/8)=1, so they are not perpendicular (perpendicular would be product -1). So the angles are not right angles, so it's a rhombus? But the initial selected option is parallelogram. Wait, maybe the order of the points is different. Wait, maybe the quadrilateral is PQRS with vertices in order P, Q, R, S, but maybe the sides are PQ, QR, RS, SP, but maybe the actual sides are PQ, QR, RS, SP, but perhaps the figure is not a rhombus. Wait, no—if all sides are equal and opposite sides are parallel, it's a rhombus. But the given options have parallelogram as selected. Wait, maybe I messed up the side lengths. Wait, let's recalculate SP: S(-4,4) to P(4,9): x difference 8, y difference 5, so distance $\sqrt{8^2 +5^2}=\sqrt{64+25}=\sqrt{89}$. Correct. RS: R(-9,-4) to S(-4,4): x difference 5, y difference 8, distance $\sqrt{5^2 +8^2}=\sqrt{89}$. Correct. QR: Q(-1,1) to R(-9,-4): x difference -8, y difference -5, distance $\sqrt{64+25}=\sqrt{89}$. Correct. PQ: P(4,9) to Q(-1,1): x difference -5, y difference -8, distance $\sqrt{25+64}=\sqrt{89}$. So all sides are equal. Then why is the answer parallelogram? Wait, maybe the problem is in the order of the vertices. Maybe the quadrilateral is not PQRS in the order I thought. Wait, maybe the sides are PQ, QR, RS, SP, but maybe the figure is actually a parallelogram but not a rhombus. Wait, no—if all sides are equal, it's a rhombus. But maybe the coordinates are misread. Wait, the point S is (-4,4), R is (-9,-4), Q is (-1,1), P is (4,9). Let's plot mentally: P(4,9) to Q(-1,1): steep line down. Q(-1,1) to R(-9,-4): line down to left. R(-9,-4) to S(-4,4): line up to right. S(-4,4) to P(4,9): line up to right. Wait, maybe the sides PQ and SR are parallel, and QR and PS are parallel. Wait, maybe I mixed up RS and SP. Wait, no—RS is from R to S, SP is from S to P. So RS: R(-9,-4) to S(-4,4): slope (4 - (-4))/(-4 - (-9)) = 8/5. SP: S(-4,4) to P(4,9): slope (9 - 4)/(4 - (-4)) = 5/8. Wait, no—wait, 9-4 is 5, 4 - (-4) is 8, so slope 5/8. Correct. So RS has slope 8/5, SP has slope 5/8. So PQ (8/5) || RS (8/5), QR (5/8) || SP (5/8). So it's a parallelogram. Now, are adjacent sides equal? PQ: length $\sqrt{(-5)^2 + (-8)^2} = \sqrt{89}$. QR: length $\sqrt{(-8)^2 + (-5)^2} = \sqrt{89}$. So PQ = QR. So in a parallelogram, if adjacent sides are equal, it's a rhombus. So why is the answer parallelogram? Wait, maybe the problem is that the angles are not 90 degrees, but rhombus doesn't require right angles. Wait, maybe the original problem's selected option is wrong, but according to the calculation, since all sides are equal and opposite sides are parallel, it should be a rhombus. But the given selected option is parallelogram. Wait, maybe I made a mistake in the slope of SP. Wait, S(-4,4) to P(4,9): y increases by 5, x increases by 8, so slope 5/8. Correct. QR: Q(-1,1) to R(-9,-4): y decreases by 5, x decreases by 8, slope (-5)/(-8) = 5/8. Correct. PQ: P(4,9) to Q(-1,1): y decreases by 8, x decreases by 5, slope (-8)/(-5) = 8/5. Correct. RS: R(-9,-4) to S(-4,4): y increases by 8, x increases by 5, slope 8/5. Correct. So opposite sides parallel (parallelogram), adjacent sides equal (so rhombus). But the options include rhombus. Wait, maybe the problem is that the user's selected option is parallelogram, but according to the math, it's a rhombus. Wait, no—wait, let's check the length of the diagonals. In a rhombus, diagonals are not equal (unless it's a square). Let's calculate diagonals PR and QS.

Diagonal PR: P(4,9) to R(-9,-4): distance $\sqrt{(-9 -4)^2 + (-4 -9)^2} = \sqrt{(-13)^2 + (-13)^2} = \sqrt{169 + 169} = \sqrt{338} = 13\sqrt{2}$

Diagonal QS: Q(-1,1) to S(-4,4): distance $\sqrt{(-4 - (-1))^2 + (4 - 1)^2} = \sqrt{(-3)^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}$

In a rhombus, diagonals are not equal (unless square), but here diagonals are $13\sqrt{2}$ and $3\sqrt{2}$, which are not equal, so it's a rhombus (since all sides equal, opposite sides parallel, diagonals not equal). But the options have parallelogram as the selected answer. Wait, maybe the initial selection is wrong, but according to the problem, the options are parallelogram, rectangle, rhombus, square. Wait, maybe the user made a mistake in the selected option, but according to the calculation, since opposite sides are parallel (parallelogram) and all sides equal (rhombus), but maybe the problem considers that the angles are not 90 degrees, but rhombus is a type of parallelogram. Wait, no—rhombus is a subset of parallelogram. So the most specific name should be rhombus. But the given selected option is parallelogram. Wait, maybe I messed up the side lengths. Wait, let's recalculate PQ: from (4,9) to (-1,1): x difference -5, y difference -8, so distance $\sqrt{25 + 64} = \sqrt{89}$. QR: from (-1,1) to (-9,-4): x difference -8, y difference -5, distance $\sqrt{64 +25} = \sqrt{89}$. So PQ = QR, so in a parallelogram, if adjacent sides are equal, it's a rhombus. So the correct answer should be rhombus. But the given selected option is parallelogram. Maybe the problem has a typo, but according to the calculation, the most specific name is rhombus. Wait, no—wait, maybe the order of the vertices is different. Maybe the quadrilateral is P(4,9), Q(-1,1), S(-4,4), R(-9,-4), so the sides are PQ, QS, SR, RP? No, the graph shows the points connected as P to Q to R to S to P. So the sides are PQ, QR, RS, SP.

Wait, maybe the key is that the slopes of adjacent sides are not negative reciprocals (so not rectangle), and all sides are equal (so rhombus). But the options include rhombus. So why is the selected option parallelogram? Maybe the original problem's answer is wrong, but according to the math, since it's a parallelogram with all sides equal, it's a rhombus. But maybe I made a mistake in the slope of SP. Wait, S(-4,4) to P(4,9): (9 - 4)/(4 - (-4)) = 5/8. Correct. QR: Q(-1,1) to R(-9,-4): (-4 -1)/(-9 - (-1)) = (-5)/(-8) = 5/8. Correct. So QR || SP, PQ || RS. So parallelogram. And since all sides are equal, it's a rhombus. So the most specific name is rhombus. But the given selected option is parallelogram. Maybe the problem is designed to have parallelogram as the answer, but according to the calculation, it's a rhombus. Wait, maybe the side lengths are not equal. Wait, let's calculate SP again: from (-4,4) to (4,9): x increases by 8, y increases by 5, so distance $\sqrt{8^2 +5^2} = \sqrt{64 +25} = \sqrt{89}$. RS: from (-9,-4) to (-4,4): x increases by 5, y increases by 8, distance $\sqrt{5^2 +8^2} = \sqrt{25 +64} = \sqrt{89}$. So SP = RS? No, SP is $\sqrt{89}$, RS is $\sqrt{89}$, so they are equal. PQ is $\sqrt{89}$, QR is $\sqrt{89}$. So all sides equal. So it's a rhombus.

But maybe the problem's answer is parallelogram, so perhaps the intended answer is parallelogram, considering that the user selected it. But according to the calculation, the most specific name is rhombus. However, maybe the initial selection is correct, and I made a mistake. Wait, no—rhombus is more specific than parallelogram. So the correct answer should be rhombus. But the given options have rhombus as an option. So the answer is rhombus. But the user's selected option is parallelogram. Maybe the problem has an error, but according to the math, the most specific name is rhombus.

Answer:

rhombus