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: Identify change in x - coordinate

To find the translation rule, observe the change in the x - coordinate of a point. For example, if we take point A, its x - coordinate changes. If we assume a general point (x,y) of triangle ABC, to get to the corresponding point of triangle A'B'C', we see that the x - coordinate of a point in ABC increases by 3. For a point (x,y) in ABC, the x - coordinate of the corresponding point in A'B'C' is x + 3.

Step2: Identify change in y - coordinate

The y - coordinate of a point in ABC decreases by 3. For a point (x,y) in ABC, the y - coordinate of the corresponding point in A'B'C' is y-3. So the translation rule is (x,y)→(x + 3,y - 3). Also, this means the triangle is translated 3 units to the right (because of the + 3 in the x - direction) and 3 units down (because of the - 3 in the y - direction).

Answer:

The rule for the translation can be written as (x,y)→(x + 3,y - 3).
The rule for the translation can be written as \(T_{3,-3}(x,y)\) (equivalent to the vector - based notation where the first number is the horizontal displacement and the second is the vertical displacement). So the correct options are:
The rule for the translation can be written as \((x,y)\to(x + 3,y - 3)\); The rule for the translation can be written as \(T_{3,-3}(x,y)\) (assuming the second option was meant to be \(T_{3,-3}(x,y)\) instead of the garbled \(T_{3 - 5}(x,y)\) in the original). If we go by the given options strictly, the correct ones are:
C. The rule for the translation can be written as \((x,y)\to(x + 3,y - 3)\)
No other completely correct option as written, but if we correct the notation in the second option to \(T_{3,-3}(x,y)\) it would be correct too.