Randy draws triangle ABC on the coordinate plane with vertices A(7, –4), B(10, 3), and C(6, 1). He then translates the figure so the coordinates of the image are A′(5, 1), B′(8, 8), and C′(4, 6). What rule did he use to draw the image?

Question

T–5, 2(x, y)
T–2, 5(x, y)
T2, –5(x, y)
T5, –2(x, y)

Answer 1

To determine the translation rule used by Randy, we need to find how each vertex of triangle ABC was transformed into its corresponding vertex in triangle A'B'C'.

Calculating the translation for each vertex:

1. For vertex A (7, -4) to A' (5, 1):

   • Change in x: \( 5 - 7 = -2 \)
   • Change in y: \( 1 - (-4) = 1 + 4 = 5 \)

2. For vertex B (10, 3) to B' (8, 8):

   • Change in x: \( 8 - 10 = -2 \)
   • Change in y: \( 8 - 3 = 5 \)

3. For vertex C (6, 1) to C' (4, 6):

   • Change in x: \( 4 - 6 = -2 \)
   • Change in y: \( 6 - 1 = 5 \)

From the calculations, we observe that:

• The change in x is consistently \(-2\) (which indicates a shift 2 units to the left).
• The change in y is consistently \(+5\) (which indicates a shift 5 units up).

Writing the translation rule:

The general form of a translation rule is \( T_{h, k}(x, y) \) which indicates a translation of \( h \) units in the x-direction and \( k \) units in the y-direction.

• Here, \(h = -2\) and \(k = 5\).

Thus, the translation can be expressed as:

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

Now, looking at the given options:

1. \( T_{-5, 2}(x, y) \) - Not correct
2. \( T_{-2, 5}(x, y) \) - This matches our rule
3. \( T_{2, -5}(x, y) \) - Not correct
4. \( T_{5, -2}(x, y) \) - Not correct

Conclusion: The correct translation rule Randy used is \( T_{-2, 5}(x, y) \).