QUESTION IMAGE
Question
which set of work correctly determines whether sean is correct?
| $qs = \sqrt{(2 + 2)^2 + (2 - 4)^2} = \sqrt{20}$
$rt = \sqrt{(-1 - 1)^2 + (1 - 5)^2} = \sqrt{20}$
therefore, qrst is a square. | $qs = \sqrt{(2 + 2)^2 + (2 - 4)^2} = \sqrt{20}$
$rt = \sqrt{(-1 - 1)^2 + (1 - 5)^2} = \sqrt{20}$
therefore, qrst is not a square. |
| $qr = \sqrt{(1 + 2)^2 + (5 - 4)^2} = \sqrt{10}$
$rs = \sqrt{(2 - 1)^2 + (2 - 5)^2} = \sqrt{10}$
therefore, qrst is a square. | $qr = \sqrt{(1 + 2)^2 + (5 - 4)^2} = \sqrt{10}$
$rs = \sqrt{(2 - 1)^2 + (2 - 5)^2} = \sqrt{10}$
therefore, qrst is not a square. |
| $qt = \sqrt{(-1 - 2)^2 + (1 - 4)^2} = \sqrt{18}$
$st = \sqrt{(2 + 1)^2 + (2 - 1)^2} = \sqrt{10}$
therefore, qrst is a square. | $qt = \sqrt{(-1 - 2)^2 + (1 - 4)^2} = \sqrt{18}$
$st = \sqrt{(2 + 1)^2 + (2 - 1)^2} = \sqrt{10}$
therefore, qrst is not a square. |
correctly checks if \( QRST \) is a square, we recall: A square has four equal side lengths and equal diagonals (or adjacent sides are perpendicular, but side - length equality is a start). We analyze each option:
Option 1 (Top - Left):
- For \( QS \): \( QS=\sqrt{(2 + 2)^{2}+(2 - 4)^{2}}=\sqrt{16 + 4}=\sqrt{20} \)
- For \( RT \): \( RT=\sqrt{(-1 - 1)^{2}+(1 - 5)^{2}}=\sqrt{4 + 16}=\sqrt{20} \)
Just checking the lengths of the diagonals (\( QS \) and \( RT \)) is not enough to confirm a square. A rectangle also has equal diagonals, so we can't conclude \( QRST \) is a square from this alone.
Option 2 (Top - Right):
- For \( QS \): \( QS=\sqrt{(2 + 2)^{2}+(2 - 4)^{2}}=\sqrt{16 + 4}=\sqrt{20} \)
- For \( RT \): \( RT=\sqrt{(-1 - 1)^{2}+(1 - 5)^{2}}=\sqrt{4 + 16}=\sqrt{20} \)
Saying \( QRST \) is not a square just because we checked diagonals is wrong. Equal diagonals are consistent with a square (or rectangle), but we need more info, but the conclusion here is incorrect.
Option 3 (Middle - Left):
- For \( QR \): \( QR=\sqrt{(1 + 2)^{2}+(5 - 4)^{2}}=\sqrt{9 + 1}=\sqrt{10} \)
- For \( RS \): \( RS=\sqrt{(2 - 1)^{2}+(2 - 5)^{2}}=\sqrt{1 + 9}=\sqrt{10} \)
Only checking two sides (\( QR \) and \( RS \)) is not sufficient. We need to check all four sides (or at least show adjacent sides are equal and perpendicular, or all four sides are equal) to confirm a square.
Option 4 (Middle - Right):
- For \( QR \): \( QR=\sqrt{(1 + 2)^{2}+(5 - 4)^{2}}=\sqrt{9 + 1}=\sqrt{10} \)
- For \( RS \): \( RS=\sqrt{(2 - 1)^{2}+(2 - 5)^{2}}=\sqrt{1 + 9}=\sqrt{10} \)
Wait, no - let's re - evaluate. Wait, the bottom - right option:
- For \( QT \): \( QT=\sqrt{(-1 - 2)^{2}+(1 - 4)^{2}}=\sqrt{9 + 9}=\sqrt{18} \)
- For \( ST \): \( ST=\sqrt{(2 + 1)^{2}+(2 - 1)^{2}}=\sqrt{9 + 1}=\sqrt{10} \)
Since \( QT
eq ST \), and in a square all sides should be equal. So if we are checking side lengths (or in this case, if we consider \( QT \) and \( ST \) as sides, their lengths are different). But let's go back to the middle - right: Wait, no, the middle - right:
Wait, the middle - right: \( QR=\sqrt{(1 + 2)^{2}+(5 - 4)^{2}}=\sqrt{10} \), \( RS=\sqrt{(2 - 1)^{2}+(2 - 5)^{2}}=\sqrt{10} \). But to be a square, all four sides must be equal. However, the bottom - right: \( QT=\sqrt{(-1 - 2)^{2}+(1 - 4)^{2}}=\sqrt{18} \), \( ST=\sqrt{(2 + 1)^{2}+(2 - 1)^{2}}=\sqrt{10} \). Since \( QT
eq ST \), the figure cannot be a square. But let's check the middle - right again. Wait, no, the correct approach is: A square has four equal sides. Let's list the vertices (we can infer the vertices of \( QRST \) from the distance formulas). Let's assume the vertices are \( Q(1,5) \), \( R(2,2) \), \( S( - 2,2) \), \( T( - 1,1) \) (by reversing the distance formula logic).
For the middle - right option:
- \( QR \): distance between \( Q(1,5) \) and \( R(2,2) \): \( \sqrt{(2 - 1)^{2}+(2 - 5)^{2}}=\sqrt{1 + 9}=\sqrt{10} \)
- \( RS \): distance between \( R(2,2) \) and \( S( - 2,2) \): \( \sqrt{( - 2 - 2)^{2}+(2 - 2)^{2}}=\sqrt{16+0} = 4\), but in the middle - right, \( RS=\sqrt{(2 - 1)^{2}+(2 - 5)^{2}}=\sqrt{10} \), which is wrong. Wait, I think I misread the points. Let's re - examine the distance formulas.
Wait, the bottom - right option:
- \( QT \): distance between \( Q \) and \( T \): \( \sqrt{( - 1 - 2)^{2}+(1 - 4)^{2}}=\sqrt{9 + 9}=\sqrt{18} \)
- \( ST \): distance between \( S \) and \( T \): \( \sqrt{(2+1)^{2}+(2 - 1)^{2}}=\sqrt{9 + 1}=\sqrt{10} \)
Since \( QT
eq ST \), the figure cannot be a square. But the middle - right: Wait, no, the correct set is the bottom…
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correctly checks if \( QRST \) is a square, we recall: A square has four equal side lengths and equal diagonals (or adjacent sides are perpendicular, but side - length equality is a start). We analyze each option:
Option 1 (Top - Left):
- For \( QS \): \( QS=\sqrt{(2 + 2)^{2}+(2 - 4)^{2}}=\sqrt{16 + 4}=\sqrt{20} \)
- For \( RT \): \( RT=\sqrt{(-1 - 1)^{2}+(1 - 5)^{2}}=\sqrt{4 + 16}=\sqrt{20} \)
Just checking the lengths of the diagonals (\( QS \) and \( RT \)) is not enough to confirm a square. A rectangle also has equal diagonals, so we can't conclude \( QRST \) is a square from this alone.
Option 2 (Top - Right):
- For \( QS \): \( QS=\sqrt{(2 + 2)^{2}+(2 - 4)^{2}}=\sqrt{16 + 4}=\sqrt{20} \)
- For \( RT \): \( RT=\sqrt{(-1 - 1)^{2}+(1 - 5)^{2}}=\sqrt{4 + 16}=\sqrt{20} \)
Saying \( QRST \) is not a square just because we checked diagonals is wrong. Equal diagonals are consistent with a square (or rectangle), but we need more info, but the conclusion here is incorrect.
Option 3 (Middle - Left):
- For \( QR \): \( QR=\sqrt{(1 + 2)^{2}+(5 - 4)^{2}}=\sqrt{9 + 1}=\sqrt{10} \)
- For \( RS \): \( RS=\sqrt{(2 - 1)^{2}+(2 - 5)^{2}}=\sqrt{1 + 9}=\sqrt{10} \)
Only checking two sides (\( QR \) and \( RS \)) is not sufficient. We need to check all four sides (or at least show adjacent sides are equal and perpendicular, or all four sides are equal) to confirm a square.
Option 4 (Middle - Right):
- For \( QR \): \( QR=\sqrt{(1 + 2)^{2}+(5 - 4)^{2}}=\sqrt{9 + 1}=\sqrt{10} \)
- For \( RS \): \( RS=\sqrt{(2 - 1)^{2}+(2 - 5)^{2}}=\sqrt{1 + 9}=\sqrt{10} \)
Wait, no - let's re - evaluate. Wait, the bottom - right option:
- For \( QT \): \( QT=\sqrt{(-1 - 2)^{2}+(1 - 4)^{2}}=\sqrt{9 + 9}=\sqrt{18} \)
- For \( ST \): \( ST=\sqrt{(2 + 1)^{2}+(2 - 1)^{2}}=\sqrt{9 + 1}=\sqrt{10} \)
Since \( QT
eq ST \), and in a square all sides should be equal. So if we are checking side lengths (or in this case, if we consider \( QT \) and \( ST \) as sides, their lengths are different). But let's go back to the middle - right: Wait, no, the middle - right:
Wait, the middle - right: \( QR=\sqrt{(1 + 2)^{2}+(5 - 4)^{2}}=\sqrt{10} \), \( RS=\sqrt{(2 - 1)^{2}+(2 - 5)^{2}}=\sqrt{10} \). But to be a square, all four sides must be equal. However, the bottom - right: \( QT=\sqrt{(-1 - 2)^{2}+(1 - 4)^{2}}=\sqrt{18} \), \( ST=\sqrt{(2 + 1)^{2}+(2 - 1)^{2}}=\sqrt{10} \). Since \( QT
eq ST \), the figure cannot be a square. But let's check the middle - right again. Wait, no, the correct approach is: A square has four equal sides. Let's list the vertices (we can infer the vertices of \( QRST \) from the distance formulas). Let's assume the vertices are \( Q(1,5) \), \( R(2,2) \), \( S( - 2,2) \), \( T( - 1,1) \) (by reversing the distance formula logic).
For the middle - right option:
- \( QR \): distance between \( Q(1,5) \) and \( R(2,2) \): \( \sqrt{(2 - 1)^{2}+(2 - 5)^{2}}=\sqrt{1 + 9}=\sqrt{10} \)
- \( RS \): distance between \( R(2,2) \) and \( S( - 2,2) \): \( \sqrt{( - 2 - 2)^{2}+(2 - 2)^{2}}=\sqrt{16+0} = 4\), but in the middle - right, \( RS=\sqrt{(2 - 1)^{2}+(2 - 5)^{2}}=\sqrt{10} \), which is wrong. Wait, I think I misread the points. Let's re - examine the distance formulas.
Wait, the bottom - right option:
- \( QT \): distance between \( Q \) and \( T \): \( \sqrt{( - 1 - 2)^{2}+(1 - 4)^{2}}=\sqrt{9 + 9}=\sqrt{18} \)
- \( ST \): distance between \( S \) and \( T \): \( \sqrt{(2+1)^{2}+(2 - 1)^{2}}=\sqrt{9 + 1}=\sqrt{10} \)
Since \( QT
eq ST \), the figure cannot be a square. But the middle - right: Wait, no, the correct set is the bottom - right? Wait, no, let's check the middle - right again. Wait, the middle - right has \( QR=\sqrt{10} \), \( RS=\sqrt{10} \), but if we check another side, say \( ST \) and \( TQ \), we need to see if they are equal. But in the bottom - right, \( QT=\sqrt{18} \) and \( ST=\sqrt{10} \), so they are not equal, so \( QRST \) is not a square.
Wait, the correct set is the bottom - right? Wait, no, let's check the first two sides. Wait, the key is: In a square, all four sides must be equal. Let's check the middle - right:
\( QR=\sqrt{(1 + 2)^{2}+(5 - 4)^{2}}=\sqrt{9 + 1}=\sqrt{10} \)
\( RS=\sqrt{(2 - 1)^{2}+(2 - 5)^{2}}=\sqrt{1 + 9}=\sqrt{10} \)
Now check \( ST \): Let's find the coordinates of \( S \) and \( T \). If \( R(1,5) \)? No, I think the points are \( Q(1,5) \), \( R(2,2) \), \( S( - 2,2) \), \( T( - 1,1) \) is wrong. Let's take the distance formula for \( QR \): \( Q(1,5) \), \( R(2,2) \): \( \sqrt{(2 - 1)^2+(2 - 5)^2}=\sqrt{1 + 9}=\sqrt{10} \)
\( RS \): \( R(2,2) \), \( S( - 2,2) \): \( \sqrt{( - 2 - 2)^2+(2 - 2)^2}=\sqrt{16 + 0}=4 \), but in the middle - right, \( RS=\sqrt{(2 - 1)^2+(2 - 5)^2}=\sqrt{10} \), so the middle - right has an error in the coordinates of \( S \).
Wait, the bottom - right:
\( QT=\sqrt{( - 1 - 2)^2+(1 - 4)^2}=\sqrt{9 + 9}=\sqrt{18} \)
\( ST=\sqrt{(2 + 1)^2+(2 - 1)^2}=\sqrt{9 + 1}=\sqrt{10} \)
Since \( QT
eq ST \), the figure cannot be a square. So the bottom - right set of work correctly concludes that \( QRST \) is not a square because it shows two sides (\( QT \) and \( ST \)) with different lengths, and in a square, all sides must be equal.
Wait, but let's check the middle - right again. Wait, maybe the vertices are \( Q(1,5) \), \( R(2,2) \), \( S(2, - 2) \), \( T(1, - 1) \)? No, this is getting too complicated. The key is that for a square, all four sides must be equal. In the bottom - right option, \( QT=\sqrt{18} \) and \( ST=\sqrt{10} \), so they are not equal, so the conclusion that \( QRST \) is not a square is correct. In the other options:
- Top - left: Only checks diagonals, not sides.
- Top - right: Wrong conclusion from diagonal check.
- Middle - left: Only checks two sides, not enough.
- Middle - right: If we assume the side - length calculation is wrong (because the distance between \( R \) and \( S \) should be different from \( QR \) if it's not a square, but in the middle - right, it's calculated as equal, which is incorrect).
So the correct set is the bottom - right one:
\( QT=\sqrt{(-1 - 2)^{2}+(1 - 4)^{2}}=\sqrt{18} \)
\( ST=\sqrt{(2 + 1)^{2}+(2 - 1)^{2}}=\sqrt{10} \)
Therefore, \( QRST \) is not a square.
So the answer is the bottom - right set of work (the last option in the table: \( QT=\sqrt{(-1 - 2)^{2}+(1 - 4)^{2}}=\sqrt{18} \), \( ST=\sqrt{(2 + 1)^{2}+(2 - 1)^{2}}=\sqrt{10} \), Therefore, \( QRST \) is not a square).