To reflect the quadrilateral QRST across the line \(y = -1\), we need to apply the reflection formula for a point. The general rule for reflecting a point \((x, y)\) across the line \(y = k\) is given by:
\[ (x', y') = (x, 2k - y) \]
where \((x', y')\) are the coordinates of the reflected point, and \((x, y)\) are the original coordinates.
In our case, the line of reflection is \(y = -1\) (thus \(k = -1\)). We will reflect each vertex of the quadrilateral.
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Reflect point Q (3, 1): \[ y' = 2(-1) - 1 = -2 - 1 = -3 \] Thus, the new coordinates of Q after reflection will be \((3, -3)\).
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Reflect point R (4, 3): \[ y' = 2(-1) - 3 = -2 - 3 = -5 \] So, the new coordinates of R after reflection will be \((4, -5)\).
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Reflect point S (5, 3): \[ y' = 2(-1) - 3 = -2 - 3 = -5 \] The new coordinates of S after reflection will be \((5, -5)\).
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Reflect point T (4, 0): \[ y' = 2(-1) - 0 = -2 - 0 = -2 \] Therefore, the new coordinates of T after reflection will be \((4, -2)\).
Now that we have reflected all the points, we can summarize the new coordinates of quadrilateral QRST after reflection over the line \(y = -1\):
- Q' = \((3, -3)\)
- R' = \((4, -5)\)
- S' = \((5, -5)\)
- T' = \((4, -2)\)
Thus, the new coordinates of the image QRST after reflection are:
- \(Q' (3, -3)\)
- \(R' (4, -5)\)
- \(S' (5, -5)\)
- \(T' (4, -2)\)
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