Question

triangle rst is translated 6 units to the left and 7 units up. the result is △rst, 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 → final coordinates r(2, -5) → r(▢, ▢) s(6, -7) → s(▢, ▢) t(1, -8) → t(▢, ▢) (b) choose the general rule below that describes the translation mapping △rst to △rst. (x, y) → (x + 7, y - 6) (x, y) → (6x, -7y) (x, y) → (-6x, 7y) (x, y) → (x + 6, y - 7) (x, y) → (-7x, 6y) (x, y) → (7x, -6y) (x, y) → (x - 6, y + 7) (x, y) → (x - 7, y + 6)

Explanation:

Step1: Find the coordinates after translation

• For a point \((x,y)\) translated \(a\) units to the left (subtract \(a\) from \(x\)) and \(b\) units up (add \(b\) to \(y\)), the rule is \((x,y)\to(x - a,y + b)\). Here \(a = 6\) and \(b=7\).
• For \(R(2,-5)\):

\(x=2,y = - 5\). After translation, \(x'=2-6=-4\), \(y'=-5 + 7=2\). So \(R'\) is \((-4,2)\).

• For \(S(6,-7)\):

\(x = 6,y=-7\). After translation, \(x'=6-6 = 0\), \(y'=-7 + 7=0\). So \(S'\) is \((0,0)\).

• For \(T(1,-8)\):

\(x = 1,y=-8\). After translation, \(x'=1-6=-5\), \(y'=-8 + 7=-1\). So \(T'\) is \((-5,-1)\).

Step2: Determine the general rule

The rule for translation \(6\) units to the left (\(x\to x-6\)) and \(7\) units up (\(y\to y + 7\)) is \((x,y)\to(x - 6,y + 7)\).

Answer:

(a) \(R(2,-5)\to R'(-4,2)\), \(S(6,-7)\to S'(0,0)\), \(T(1,-8)\to T'(-5,-1)\)
(b) \((x,y)\to(x - 6,y + 7)\)