a line segment in a coordinate plane with a length of 76 units was rotated 60 deg about the origin and then translate 10 units to the left. what would be a resulting length of the segment after these transformation
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To find the resulting length of the line segment after the given transformations, we need to follow these steps:
1. Find the coordinates of the endpoints of the original line segment.
2. Apply the rotation transformation.
3. Apply the translation transformation.
4. Calculate the length of the resulting line segment using the coordinates of the transformed endpoints.
Let's go through these steps one by one.
Step 1: Find the coordinates of the endpoints of the original line segment.
Since the line segment is not described in the question, we can assume its coordinates based on the given information. Let's say one endpoint is at (0, 0), and the other endpoint is at (0, 76).
Step 2: Apply the rotation transformation.
To rotate a point (x, y) by 60 degrees counterclockwise about the origin, we can use the following formulas:
x' = x * cos(theta) - y * sin(theta)
y' = x * sin(theta) + y * cos(theta)
In our case, x = 0 and y = 76, and theta = 60 degrees.
x' = 0 * cos(60) - 76 * sin(60) = -38 * sqrt(3)
y' = 0 * sin(60) + 76 * cos(60) = 38
So, the transformed coordinates of the endpoints after rotation are (-38 * sqrt(3), 38) and (0, 0).
Step 3: Apply the translation transformation.
To translate a point (x, y) to the left by 10 units, we subtract 10 from the x-coordinate.
For the endpoints, the new coordinates after translation are:
(-38 * sqrt(3) - 10, 38) and (0 - 10, 0) which simplifies to (-38 * sqrt(3) - 10, 38) and (-10, 0).
Step 4: Calculate the length of the resulting line segment.
The length of a line segment can be calculated using the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's calculate the length using the coordinates of the transformed endpoints:
d = sqrt((-38 * sqrt(3) - 10 - (-10))^2 + (38 - 0)^2)
= sqrt((-38 * sqrt(3) - 20)^2 + 1444)
= sqrt(1444 + 144 * sqrt(3) + 1140 + 400)
= sqrt(2984 + 144 * sqrt(3))
Therefore, the resulting length of the line segment after the transformations is approximately sqrt(2984 + 144 * sqrt(3)) units.