If an angle measuring 30 degrees is translated 8 units left and 4 units up, what does the angle measure after the translation?
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To find the new angle measure after the translation, we need to understand what a translation is. A translation is a type of transformation that moves a shape or an object from one position to another without changing its size or shape. In this case, the angle is being translated 8 units to the left and 4 units up.
To visualize this, imagine a coordinate plane where the angle is located. Let's say the vertex of the angle is at the origin (0,0). We can place one side of the angle along the positive x-axis to simplify the calculation.
Now, if we move the angle 8 units to the left, we subtract 8 from the x-coordinate of each point on the angle. So, the new x-coordinate will be -8. Similarly, if we move the angle 4 units up, we add 4 to the y-coordinate of each point on the angle. Therefore, the new y-coordinate will be 4.
Now, we have the new coordinates of the angle: (-8, 4).
To find the angle measure after the translation, we can use basic trigonometry. Since the initial angle measures 30 degrees, it means that the ratio of the opposite side (4) to the adjacent side (-8) is equal to the tangent of the angle.
So, the tangent of the angle is given by:
tan(angle) = opposite/adjacent
tan(angle) = 4/(-8)
To find the angle, we can take the arctan (inverse tangent) of both sides:
angle = arctan(4/(-8))
Using a calculator, we find:
angle ≈ -26.57 degrees
Therefore, the angle measures approximately -26.57 degrees after the translation. Note that the negative sign indicates that the angle is in the 4th quadrant.