Let's solve the problem step-by-step.
First, we should analyze the given conditions about the triangle [tex]\(ABC\)[/tex]:
1. [tex]\(\triangle ABC\)[/tex] is an isosceles right triangle.
2. The slope of [tex]\(\overline{AB}\)[/tex] is -1.
3. [tex]\(\angle ABC = 90^\circ\)[/tex].
To simplify calculations, let's place point [tex]\(A\)[/tex] at the origin [tex]\((0,0)\)[/tex]. Let point [tex]\(B(x, y)\)[/tex]. Given that [tex]\(AB\)[/tex] has a slope of -1, the coordinates of [tex]\(B\)[/tex] can be related by the equation of slope:
[tex]\[ \text{slope of } AB = \frac{y - 0}{x - 0} = -1 \][/tex]
So, [tex]\(y = -x\)[/tex]. Therefore, [tex]\(B\)[/tex] can be represented as [tex]\((x, -x)\)[/tex].
Since [tex]\(\triangle ABC\)[/tex] is an isosceles right triangle with [tex]\(\angle ABC = 90^\circ\)[/tex], point [tex]\(C\)[/tex] can be found by rotating point [tex]\(B\)[/tex] 90 degrees counterclockwise around point [tex]\(A\)[/tex].
Thus, if [tex]\(B(x, -x)\)[/tex], then [tex]\(C\)[/tex] will be [tex]\((x, x)\)[/tex].
Next, we apply the dilation by a factor of 1.8 with the center at the origin. Point coordinates after dilation are found by multiplying each coordinate by 1.8:
For point [tex]\(A (0, 0)\)[/tex]:
[tex]\[ A' = (0 \times 1.8, 0 \times 1.8) = (0, 0) \][/tex]
For point [tex]\(B (x, -x)\)[/tex]:
[tex]\[ B' = (1.8x, -1.8x) \][/tex]
For point [tex]\(C (x, x)\)[/tex]:
[tex]\[ C' = (1.8x, 1.8x) \][/tex]
We now find the slope of [tex]\(\overline{B'C'}\)[/tex]:
[tex]\[ B' = (1.8x, -1.8x) \][/tex]
[tex]\[ C' = (1.8x, 1.8x) \][/tex]
The slope of [tex]\(\overline{B'C'}\)[/tex] is given by:
[tex]\[ \text{slope of } B'C' = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1.8x - (-1.8x)}{1.8x - 1.8x} = \frac{1.8x + 1.8x}{0} \][/tex]
Since we are dividing by zero (the x-coordinates are the same), this means the line is vertical, and the slope is undefined.
From the given options:
A. -1
B. 0
C. 1
D. 1.5
E. 2
In this context, the closest match to an undefined vertical slope (which would be infinite) is consideration of the slope being essentially vertical, making it an undefined solution that corresponds to the problem scenario.
Since none of the given choices accurately represent an undefined slope, reviewing the options might have been necessary if considering horizontal where uniform functionality in a given problem happens often or any typographical mishap.
Hence, given choices need to be revised considering undefined slopes, and typically matched as closest mathematical implications provide.
Therefore, with computed adjustments not precisely stated, mathematically it derived but otherwise might be adaptable in context resources available fitting realistic computational:
The correct selection considering looks incorrectly editable scenario (insisted options actual).
[Or]undefined typical noted nearest computation pointed implies broad selection nearest adaptability.
