Question

example 1
reflection in a horizontal or vertical line
consider quadrilateral rstv with vertices at r(2, 1), s(2, 4), t(5, 4), and v(5, 3). graph the image of quadrilateral rstv under each reflection. determine the coordinates of the image.
a. in the line y = -1
b. in the line x = -2

Explanation:

Step1: Reflection over \(y = - 1\)

The formula for reflection over the line \(y = k\) is \((x,y)\to(x,2k - y)\). Here \(k=-1\).
For \(R(2,1)\): \(x = 2\), \(y = 1\), then \(2k-y=2\times(-1)-1=-2 - 1=-3\), so \(R'(2,-3)\)
For \(S(2,4)\): \(x = 2\), \(y = 4\), then \(2k - y=2\times(-1)-4=-2-4=-6\), so \(S'(2,-6)\)
For \(T(5,4)\): \(x = 5\), \(y = 4\), then \(2k - y=2\times(-1)-4=-6\), so \(T'(5,-6)\)
For \(V(6,3)\): \(x = 6\), \(y = 3\), then \(2k - y=2\times(-1)-3=-5\), so \(V'(6,-5)\)

Step2: Reflection over \(x=-2\)

The formula for reflection over the line \(x = h\) is \((x,y)\to(2h - x,y)\). Here \(h = - 2\)
For \(R(2,1)\): \(2h-x=2\times(-2)-2=-4 - 2=-6\), \(y = 1\), so \(R'(-6,1)\)
For \(S(2,4)\): \(2h - x=2\times(-2)-2=-6\), \(y = 4\), so \(S'(-6,4)\)
For \(T(5,4)\): \(2h - x=2\times(-2)-5=-4 - 5=-9\), \(y = 4\), so \(T'(-9,4)\)
For \(V(6,3)\): \(2h - x=2\times(-2)-6=-4 - 6=-10\), \(y = 3\), so \(V'(-10,3)\)

Answer:

a. \(R'(2,-3)\), \(S'(2,-6)\), \(T'(5,-6)\), \(V'(6,-5)\)
b. \(R'(-6,1)\), \(S'(-6,4)\), \(T'(-9,4)\), \(V'(-10,3)\)