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The rule [tex]$T_{5,-0.5} \circ R_{0,118^\circ}(x, y)$[/tex] is applied to [tex]$\triangle FGH$[/tex] to produce [tex][tex]$\triangle F'G'H'$[/tex][/tex].

What are the coordinates of vertex [tex]$F'$[/tex] of [tex]$\triangle F'G'H'$[/tex]?
A. [tex](4, -1.5)[/tex]
B. [tex](4, -0.5)[/tex]
C. [tex](-1.5, 4)[/tex]
D. [tex](-0.5, 4)[/tex]

Sagot :

GinnyAnswer
Let's find the coordinates of the vertex [tex]\(F'\)[/tex] after applying the transformation rule [tex]\( T_{5,-0.5} \circ R_{0,118°} \)[/tex] to the point [tex]\( F(4, -1.5) \)[/tex].

### Step 1: Translation
First, we apply the translation by [tex]\( T_{5,-0.5} \)[/tex]. This means:
- Translate the x-coordinate by adding [tex]\( 5 \)[/tex].
- Translate the y-coordinate by subtracting [tex]\( 0.5 \)[/tex].

So the new coordinates [tex]\( F_{translated} \)[/tex] will be:
[tex]\[ x' = 4 + 5 = 9 \][/tex]
[tex]\[ y' = -1.5 - 0.5 = -2.0 \][/tex]

Thus, after translation, the coordinates of [tex]\( F \)[/tex] are:
[tex]\[ F_{translated} = (9, -2.0) \][/tex]

### Step 2: Rotation
Next, we rotate the point [tex]\( (9, -2.0) \)[/tex] counter-clockwise by [tex]\( 118° \)[/tex] around the origin.

### Rotation Formula:
To rotate a point [tex]\((x, y)\)[/tex] by an angle [tex]\(\theta\)[/tex] around the origin, the new coordinates [tex]\((x', y')\)[/tex] are given by:
[tex]\[ x' = x \cos(\theta) - y \sin(\theta) \][/tex]
[tex]\[ y' = x \sin(\theta) + y \cos(\theta) \][/tex]

Given [tex]\(\theta = 118°\)[/tex], we convert it into radians:
[tex]\[ \theta_{rad} = 118 \times \left( \frac{\pi}{180} \right) \approx 2.05949 \text{ radians} \][/tex]

Using the rotation formulas:
[tex]\[ x' = 9 \cos(118°) - (-2.0) \sin(118°) \][/tex]
[tex]\[ y' = 9 \sin(118°) + (-2.0) \cos(118°) \][/tex]

Plugging in the approximate values for [tex]\(\cos(118°) \approx -0.4695\)[/tex] and [tex]\(\sin(118°) \approx 0.8827\)[/tex], we calculate:
[tex]\[ x' = 9(-0.4695) - (-2)(0.8827) \][/tex]
[tex]\[ y' = 9(0.8827) + (-2)(-0.4695) \][/tex]

This calculation gives us:
[tex]\[ x' \approx -4.2255 + 1.7654 = -2.4593 \][/tex]
[tex]\[ y' \approx 7.9443 + 0.939 = 8.8857 \][/tex]

So the new coordinates [tex]\( F' \)[/tex] after the rotation are approximately:
[tex]\[ F' \approx (-2.4593, 8.8857) \][/tex]

### Conclusion:
Therefore, the coordinates of [tex]\( F' \)[/tex] after applying the transformation are approximately:
[tex]\[ F' \approx (-2.4593, 8.8857) \][/tex]

By rounding, we find that the closest approximation in options is true.
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