Question

1. graph \\( \triangle a b c \\) with vertices \\( a ( 2,1 ), b ( - 1,2 ) \\) and \\( c ( - 2, - 1 ) \\) and its image after the following sequence of transformations.

translation: \\( ( x, y ) \
ightarrow ( x - 1, y - 3 ) \\)
translation: \\( ( x, y ) \
ightarrow ( x + 4, y + 3 ) \\)
write a rule for the translation that maps each figure to its image.
2.
3.

1. if \\( \triangle x y z \\) is the image of a translation of \\( \triangle a b c, a \\) is at \\( ( 1,1 ) \\) and \\( x \\) is at \\( ( 5, - 2 ) \\), what translation rule maps \\( \triangle a b c \\) to \\( \triangle x y z \\) ?

a. \\( ( x, y ) \
ightarrow ( x + 4, y - 3 ) \\)
b. \\( ( x, y ) \
ightarrow ( x - 4, y - 3 ) \\)
c. \\( ( x, y ) \
ightarrow ( x + 4, y + 2 ) \\)
d. \\( ( x, y ) \
ightarrow ( x + 4, y - 2 ) \\)

1. triangle \\( a b c \\) with coordinates \\( a ( 3, - 2 ), b ( 3,4 ) \\), and \\( c ( 6,5 ) \\) it is translated following the rule \\( ( x, y ) \

ightarrow ( x + 3, y - 8 ) \\). what will be the resulting coordinates of the vertices of the transformed triangle \\( a ^ { prime }, b ^ { prime } \\), \\( c ^ { prime } \\) ?
a. \\( a ( 6, - 10 ), b ( 6, - 4 ) \\), and \\( c ( 9, - 3 ) \\)
b. \\( a ( 6,2 ), b ( 3,12 ) \\), and \\( c ( 9,13 ) \\)
c. \\( a ( 0, - 10 ), b ( 0, - 4 ) \\), and \\( c ( 3, - 3 ) \\)
d. \\( a ( 0,2 ), b ( 0,12 ) \\), and \\( c ( 3,13 ) \\)

Explanation:

Step1: Calculate the translation rule for question 4

For a translation from point \(A(1,1)\) to \(X(5, - 2)\), the \(x\)-coordinate change is \(5 - 1=4\), and the \(y\)-coordinate change is \(-2 - 1=-3\).
The translation rule is \((x,y)\to(x + 4,y-3)\)

Step2: Calculate the coordinates for question 5

For point \(A(3,-2)\):
\(x\)-coordinate: \(3+3 = 6\), \(y\)-coordinate: \(-2-8=-10\), so \(A'(6,-10)\)
For point \(B(3,4)\):
\(x\)-coordinate: \(3 + 3=6\), \(y\)-coordinate: \(4-8=-4\), so \(B'(6,-4)\)
For point \(C(6,5)\):
\(x\)-coordinate: \(6+3 = 9\), \(y\)-coordinate: \(5-8=-3\), so \(C'(9,-3)\)

Answer:

1. A. \((x,y)\to(x + 4,y-3)\)
2. A. \(A(6,-10),B(6,-4)\), and \(C(9,-3)\)