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Randy draws triangle [tex]$ABC$[/tex] on the coordinate plane with vertices [tex]$A (7,-4)$[/tex], [tex]$B (10,3)$[/tex], and [tex]$C(6,1)$[/tex]. He then translates the figure so the coordinates of the image are [tex]$A ^{\prime}(5,1)$[/tex], [tex]$B ^{\prime}(8,8)$[/tex], and [tex]$C ^{\prime}(4,6)$[/tex]. What rule did he use to draw the image?

A. [tex]$T_{-5,2}(x, y)$[/tex]
B. [tex]$T_{-2,5}(x, y)$[/tex]
C. [tex]$T_{2,-5}(x, y)$[/tex]
D. [tex]$T_{5,-2}(x, y)$[/tex]



Answer :

GinnyAnswer
Let's find the translation rule Randy used to transform triangle [tex]\(ABC\)[/tex] to triangle [tex]\(A'B'C'\)[/tex]. The translation rule will be the same for all points on the triangle, so we can determine it by comparing the coordinates of one pair of corresponding points. Let's use point [tex]\(A\)[/tex] and its image [tex]\(A'\)[/tex].

1. The initial coordinates of [tex]\(A\)[/tex] are [tex]\((7, -4)\)[/tex].
2. The coordinates of the translated point [tex]\(A'\)[/tex] are [tex]\((5, 1)\)[/tex].

To find the translation rule, we determine how each coordinate of [tex]\(A\)[/tex] was changed to get [tex]\(A'\)[/tex]:

Step-by-step Calculation:

1. Horizontal Translation:

The x-coordinate of [tex]\(A\)[/tex] is 7. The x-coordinate of [tex]\(A'\)[/tex] is 5. The change in the x-coordinate is:

[tex]\[ T_x = 5 - 7 = -2 \][/tex]

This means that the x-coordinate was translated by [tex]\(-2\)[/tex].

2. Vertical Translation:

The y-coordinate of [tex]\(A\)[/tex] is [tex]\(-4\)[/tex]. The y-coordinate of [tex]\(A'\)[/tex] is 1. The change in the y-coordinate is:

[tex]\[ T_y = 1 - (-4) = 1 + 4 = 5 \][/tex]

This means that the y-coordinate was translated by [tex]\(5\)[/tex].

Translation Rule:

The translation that Randy used moves each point by [tex]\(-2\)[/tex] units in the x-direction and [tex]\(5\)[/tex] units in the y-direction. This can be written as:

[tex]\[ T_{-2, 5}(x, y) \][/tex]

To verify this, let's apply the same rule to points [tex]\(B\)[/tex] and [tex]\(C\)[/tex] and check if they translate to [tex]\(B'\)[/tex] and [tex]\(C'\)[/tex]:

- For [tex]\(B (10,3) \rightarrow B' (8,8)\)[/tex]:
- x-coordinate: [tex]\(8 - 10 = -2\)[/tex]
- y-coordinate: [tex]\(8 - 3 = 5\)[/tex]

- For [tex]\(C (6,1) \rightarrow C' (4,6)\)[/tex]:
- x-coordinate: [tex]\(4 - 6 = -2\)[/tex]
- y-coordinate: [tex]\(6 - 1 = 5\)[/tex]

Both calculations confirm the translation rule [tex]\( T_{-2, 5}(x, y) \)[/tex].

Therefore, the rule that Randy used to draw the image is:

[tex]\[ \boxed{T_{-2,5}(x, y)} \][/tex]

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