Question

triangle rst is translated 5 units to the right and 7 units down. the result is \\( \triangle r ^ { prime } s ^ { prime } t ^ { prime } \\), as shown below. (a) the arrows below show that the coordinates on the left are mapped to the coordinates on the right. fill in the blanks to give the coordinates after the translation. original coordinates \\( \
ightarrow \\) final coordinates \\( r ( - 4,4 ) \
ightarrow r ^ { prime } ( square, square ) \\) \\( s ( - 7,5 ) \
ightarrow s ^ { prime } ( square, square ) \\) \\( t ( - 8,8 ) \
ightarrow t ^ { prime } ( square, square ) \\) (b) choose the general rule below that describes the translation mapping \\( \triangle r s t \\) to \\( \triangle r ^ { prime } s ^ { prime } t ^ { prime } \\). \\( ( x, y ) \
ightarrow ( x - 7, y + 5 ) \\) \\( ( x, y ) \
ightarrow ( x + 7, y - 5 ) \\) \\( ( x, y ) \
ightarrow ( 7 x, - 5 y ) \\) \\( ( x, y ) \
ightarrow ( - 7 x, 5 y ) \\) \\( ( x, y ) \
ightarrow ( - 5 x, 7 y ) \\) \\( ( x, y ) \
ightarrow ( 5 x, - 7 y ) \\) \\( ( x, y ) \
ightarrow ( x - 5, y + 7 ) \\) \\( ( x, y ) \
ightarrow ( x + 5, y - 7 ) \\)

Explanation:

Step1: Calculate \( R' \)

For a point \((x,y)\) translated \(5\) units right (add \(5\) to \(x\)) and \(7\) units down (subtract \(7\) from \(y\)).
For \(R(-4,4)\):
\(x=-4 + 5=1\), \(y = 4-7=-3\). So \(R'(1,-3)\)

Step2: Calculate \( S' \)

For \(S(-7,5)\):
\(x=-7 + 5=-2\), \(y = 5-7=-2\). So \(S'(-2,-2)\)

Step3: Calculate \( T' \)

For \(T(-8,8)\):
\(x=-8 + 5=-3\), \(y = 8-7 = 1\). So \(T'(-3,1)\)

Step4: Determine the general rule

The rule for translation is \((x,y)\to(x + 5,y-7)\)

Answer:

a) \(R'(1,-3)\), \(S'(-2,-2)\), \(T'(-3,1)\)
b) \((x,y)\to(x + 5,y-7)\)