Question

1. the coordinates of △abc are a(2,3), b(6,6), and c(7,2). after a translation, the image of vertex a is a(-6,1).

a) give the coordinates of b and c after the same translation.
b) graph △abc and △abc on the graph above and to the right.

1. the coordinates of defg are d(2,-1), e(5,-2), f(4,-5), and g(1,-4).

a) describe in coordinate mapping notation a translation that will move vertex e to the origin.
(x,y)→(__,__)
b) give the coordinates of d, e, f, and g after the translation described in part a).

Explanation:

Step1: Determine the translation rule for problem 6a

To find the translation rule from \(A(2,3)\) to \(A'(-6,1)\), we calculate the change in \(x\) and \(y\) - coordinates.
The change in \(x\) - coordinate: \(\Delta x=-6 - 2=-8\).
The change in \(y\) - coordinate: \(\Delta y=1 - 3=-2\).
The translation rule is \((x,y)\to(x - 8,y - 2)\).
For \(B(6,6)\):
\(x\) - coordinate of \(B'\): \(6-8=-2\)
\(y\) - coordinate of \(B'\): \(6 - 2 = 4\)
For \(C(7,2)\):
\(x\) - coordinate of \(C'\): \(7-8=-1\)
\(y\) - coordinate of \(C'\): \(2 - 2=0\)

Step2: Determine the translation rule for problem 7a

We want to move \(E(5,-2)\) to \((0,0)\).
The change in \(x\) - coordinate: \(\Delta x=0 - 5=-5\).
The change in \(y\) - coordinate: \(\Delta y=0-(-2)=2\).
The translation rule is \((x,y)\to(x - 5,y + 2)\)

Step3: Apply the translation rule for problem 7b

For \(D(2,-1)\):
\(x\) - coordinate of \(D'\): \(2-5=-3\)
\(y\) - coordinate of \(D'\): \(-1 + 2=1\)
For \(E(5,-2)\):
\(x\) - coordinate of \(E'\): \(5-5 = 0\)
\(y\) - coordinate of \(E'\): \(-2+2=0\)
For \(F(4,-5)\):
\(x\) - coordinate of \(F'\): \(4-5=-1\)
\(y\) - coordinate of \(F'\): \(-5 + 2=-3\)
For \(G(1,-4)\):
\(x\) - coordinate of \(G'\): \(1-5=-4\)
\(y\) - coordinate of \(G'\): \(-4 + 2=-2\)

Answer:

6a. \(B'(-2,4)\), \(C'(-1,0)\)
7a. \((x,y)\to(x - 5,y + 2)\)
7b. \(D'(-3,1)\), \(E'(0,0)\), \(F'(-1,-3)\), \(G'(-4,-2)\)