quadrilateral QRST with the coordinates Q (3, 1), R (4, 3), S (5, 3), and T (4, 0) is reflected over the line y = -1 what are the new coordinates of the image QRST. show your method

Question

GPT-4o mini · Answer

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:

$(x^{'}, y^{'}) = (x, 2 k - y)$

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.

Reflect point Q (3, 1): $y^{'} = 2 (- 1) - 1 = - 2 - 1 = - 3$ Thus, the new coordinates of Q after reflection will be $(3, - 3)$ .
Reflect point R (4, 3): $y^{'} = 2 (- 1) - 3 = - 2 - 3 = - 5$ So, the new coordinates of R after reflection will be $(4, - 5)$ .
Reflect point S (5, 3): $y^{'} = 2 (- 1) - 3 = - 2 - 3 = - 5$ The new coordinates of S after reflection will be $(5, - 5)$ .
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:

$Q^{'} (3, - 3)$
$R^{'} (4, - 5)$
$S^{'} (5, - 5)$
$T^{'} (4, - 2)$