Question

which are side lengths of quadrilateral pqrs? check all that apply. 4 units 5 units 6 units \\( \sqrt { 13 } \\) units \\( 4 \sqrt { 2 } \\) units

Explanation:

Step1: Calculate PQ length

Points \( P(0, 4) \) and \( Q(2, 4) \). Since y - coordinates are same, distance is \( |2 - 0| = 2 \)? Wait, no, wait: \( PQ \): \( x \)-difference is \( 2 - 0 = 2 \), \( y \)-difference \( 4 - 4 = 0 \). So length \( \sqrt{(2 - 0)^2+(4 - 4)^2}=\sqrt{4 + 0}=2 \)? Wait, no, the options have 4? Wait, maybe I misread. Wait, \( P(0,4) \), \( Q(2,4) \): horizontal distance, so \( 2 - 0 = 2 \)? No, wait the grid: each square is 1 unit. Wait, \( P(0,4) \) to \( Q(2,4) \): that's 2 units? But option has 4. Wait, maybe \( QR \)? \( Q(2,4) \) to \( R(2,-2) \): vertical distance, \( 4 - (-2)=6 \). So \( QR = 6 \) units. Then \( P(0,4) \) to \( S(-2,1) \): distance formula: \( \sqrt{(0 - (-2))^2+(4 - 1)^2}=\sqrt{4 + 9}=\sqrt{13} \). \( S(-2,1) \) to \( R(2,-2) \): \( \sqrt{( - 2 - 2)^2+(1 - (-2))^2}=\sqrt{16 + 9}=\sqrt{25}=5 \). \( PQ \): \( \sqrt{(2 - 0)^2+(4 - 4)^2}=2 \)? No, wait maybe I messed up points. Wait the quadrilateral is PQRS, so order is P - Q - R - S - P? Let's check each side:

1. \( PQ \): \( P(0,4) \), \( Q(2,4) \): horizontal line, length \( 2 - 0 = 2 \)? No, that's 2, but option has 4? Wait maybe the grid is different. Wait, maybe \( P(0,4) \) to \( Q(2,4) \): 2 units? But option 4. Wait, maybe \( PS \)? No, let's recalculate all sides:

- \( PQ \): \( P(0,4) \), \( Q(2,4) \): \( \sqrt{(2 - 0)^2 + (4 - 4)^2} = 2 \) (not in options? No, options have 4,5,6,√13,4√2. Wait maybe I got the points wrong. Wait \( S(-2,1) \), \( P(0,4) \): distance \( \sqrt{(0 - (-2))^2 + (4 - 1)^2} = \sqrt{4 + 9} = \sqrt{13} \). \( Q(2,4) \) to \( R(2,-2) \): vertical distance, \( 4 - (-2) = 6 \), so \( QR = 6 \) units. \( R(2,-2) \) to \( S(-2,1) \): \( \sqrt{(2 - (-2))^2 + (-2 - 1)^2} = \sqrt{16 + 9} = 5 \) units. \( P(0,4) \) to \( Q(2,4) \): 2 units? No, but option 4. Wait, maybe \( P(0,4) \) to \( S(-2,1) \): no, \( \sqrt{13} \) is there. Wait, let's list all sides:

Quadrilateral PQRS: vertices in order P(0,4), Q(2,4), R(2,-2), S(-2,1), back to P(0,4).

So sides:

1. PQ: between (0,4) and (2,4): length \( |2 - 0| = 2 \) (since y same). But option 4? No, maybe I misread the points. Wait, maybe P is (0,4), Q is (4,4)? No, the graph shows Q at (2,4). Wait, maybe the grid is 2 units per square? No, the axes are labeled with 1 unit per grid. Wait, maybe the problem is PQRS with sides PQ, QR, RS, SP.

- PQ: (0,4) to (2,4): 2 units (not in options? No, options have 4,5,6,√13,4√2. Wait, maybe SP: S(-2,1) to P(0,4): \( \sqrt{(0 - (-2))^2 + (4 - 1)^2} = \sqrt{4 + 9} = \sqrt{13} \) (option √13).

- QR: (2,4) to (2,-2): vertical distance, \( 4 - (-2) = 6 \) (option 6 units).

- RS: (2,-2) to (-2,1): \( \sqrt{(2 - (-2))^2 + (-2 - 1)^2} = \sqrt{16 + 9} = 5 \) (option 5 units).

- SP: (-2,1) to (0,4): \( \sqrt{(0 - (-2))^2 + (4 - 1)^2} = \sqrt{13} \) (option √13).

Wait, but also PQ: (0,4) to (2,4) is 2, but option 4? No, maybe I made a mistake. Wait, maybe the quadrilateral is P(0,4), Q(2,4), R(2,-2), S(-2,1), so PQ is 2, QR is 6, RS is 5, SP is √13. So the options that apply are 5 units (RS), 6 units (QR), √13 units (SP). Wait, but also, is there a side of 4 units? Wait, maybe I misread the points. Wait, P(0,4) to Q(2,4): 2 units. No. Wait, maybe S(-2,1) to R(2,-2) is 5, Q(2,4) to R(2,-2) is 6, S(-2,1) to P(0,4) is √13. So the correct options are 5 units, 6 units, √13 units. Wait, let's recheck:

- PQ: distance between (0,4) and (2,4): \( \sqrt{(2-0)^2 + (4-4)^2} = 2 \) (not in options).

- QR: (2,4) to (2,-2): \( \sqrt{(2-2)^2 + (-2-4)^2} = \sqrt{0 + 36} = 6 \) (so 6 units, option present).

- RS: (2,-2) to (-2,1): \( \sqrt{(-2-2)^2 + (1 - (-2))^2} = \sqrt{16 + 9} = 5 \) (5 units, option present).

- SP: (-2,1) to (0,4): \( \sqrt{(0 - (-2))^2 + (4 - 1)^2} = \sqrt{4 + 9} = \sqrt{13} \) (√13 units, option present).

So the correct options are 5 units, 6 units, √13 units. Wait, but the options are: 4 units, 5 units, 6 units, √13 units, 4√2 units. So 5, 6, √13.

Step1: Calculate PQ length (not needed, as it's 2, not in options)

Step2: Calculate QR length

Points \( Q(2,4) \) and \( R(2,-2) \). Distance formula: \( \sqrt{(2 - 2)^2+(-2 - 4)^2}=\sqrt{0 + 36}=6 \). So \( QR = 6 \) units.

Step3: Calculate RS length

Points \( R(2,-2) \) and \( S(-2,1) \). Distance formula: \( \sqrt{(-2 - 2)^2+(1 - (-2))^2}=\sqrt{16 + 9}=\sqrt{25}=5 \). So \( RS = 5 \) units.

Step4: Calculate SP length

Points \( S(-2,1) \) and \( P(0,4) \). Distance formula: \( \sqrt{(0 - (-2))^2+(4 - 1)^2}=\sqrt{4 + 9}=\sqrt{13} \). So \( SP = \sqrt{13} \) units.

Answer:

5 units, 6 units, \( \sqrt{13} \) units (corresponding to the options: 5 units, 6 units, \( \sqrt{13} \) units)