Question

which statements are true about triangle abc and its translated image, abc? select two options.
the rule for the translation can be written as t_ - 5,3(x,y).
the rule for the translation can be written as t_3, - 5(x,y).
the rule for the translation can be written as (x,y)→(x + 3,y - 3).
the rule for the translation can be written as (x,y)→(x - 3,y - 3).
triangle abc has been translated 3 units to the right and 5 units down.

Explanation:

Step1: Analyze horizontal translation

To find the horizontal translation, compare the x - coordinates of corresponding points. For example, if we take point A and A'. Let's assume A has coordinates \((x_1,y_1)\) and A' has coordinates \((x_2,y_2)\). By observing the graph, we see that the x - coordinate of the points of the translated triangle is 3 units more than the original triangle's points. So the horizontal translation is \(x
ightarrow x + 3\).

Step2: Analyze vertical translation

To find the vertical translation, compare the y - coordinates of corresponding points. The y - coordinate of the points of the translated triangle is 3 units less than the original triangle's points. So the vertical translation is \(y
ightarrow y- 3\).

Step3: Determine translation rule

The rule for translation in coordinate - geometry is \((x,y)
ightarrow(x + a,y + b)\), where \(a\) is the horizontal displacement and \(b\) is the vertical displacement. Here \(a = 3\) and \(b=-3\), so the rule is \((x,y)
ightarrow(x + 3,y - 3)\). Also, in the vector - based notation for translation \(T_{a,b}(x,y)=(x + a,y + b)\), here \(a = 3\) and \(b=-3\), so it can be written as \(T_{3,-3}(x,y)\).

Answer:

The rule for the translation can be written as \(T_{3,-3}(x,y)\); The rule for the translation can be written as \((x,y)
ightarrow(x + 3,y - 3)\)