1. Demonstrate the translation of a line segment of 3 units to the left and 5 units down if the endpoints of the line segment are (-2, 7) and (0,3).
2. If a line segment with endpoints (6,3) and (6,7) is translated 3 units up and 4 units to the right, how long is the line segment after the translation?
3. A line segment has endpoints (1,1) and (4,4). Demonstrating the properties of reflections across the y-axis, what are the reflected endpoints?
4. A line contains the points (6,-3) and (3,1). When this line is reflected across the y-axis, what are the two new points?
To reflect a point across the x-axis, we keep the y-coordinate the same and change the sign of the x-coordinate.
The initial endpoints are (0, 5) and (6, 5).
To reflect across the x-axis, we change the sign of the y-coordinates:
Reflected endpoints: (0, -5) and (6, -5)
The length of the line segment after reflection can be found using the distance formula:
Length = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((6 - 0)^2 + (-5 - (-5))^2)
= sqrt(6^2 + 0)
= sqrt(36 + 0)
= sqrt(36)
= 6
So, the length of the line segment after reflecting across the x-axis remains unchanged at 6 units.
1. To translate the line segment 3 units to the left and 5 units down, we subtract 3 from the x-coordinates and subtract 5 from the y-coordinates of the endpoints.
The initial endpoints are (-2, 7) and (0, 3).
To translate 3 units to the left, we subtract 3 from both x-coordinates:
New endpoints: (-2 - 3, 7) and (0 - 3, 3)
= (-5, 7) and (-3, 3)
To translate 5 units down, we subtract 5 from both y-coordinates:
New endpoints: (-5, 7 - 5) and (-3, 3 - 5)
= (-5, 2) and (-3, -2)
So, the translated line segment has endpoints (-5, 2) and (-3, -2).
2. To translate the line segment 3 units up and 4 units to the right, we add 3 to the y-coordinates and add 4 to the x-coordinates of the endpoints.
The initial endpoints are (6, 3) and (6, 7).
To translate 3 units up, we add 3 to both y-coordinates:
New endpoints: (6, 3 + 3) and (6, 7 + 3)
= (6, 6) and (6, 10)
To translate 4 units to the right, we add 4 to both x-coordinates:
New endpoints: (6 + 4, 6) and (6 + 4, 10)
= (10, 6) and (10, 10)
The length of the line segment after the translation can be found using the distance formula:
Length = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((10 - 10)^2 + (10 - 6)^2)
= sqrt(0 + 16)
= sqrt(16)
= 4
So, the length of the line segment after the translation is 4 units.
3. To reflect a point across the y-axis, we change the sign of its x-coordinate while keeping the y-coordinate the same.
The initial endpoints are (1, 1) and (4, 4).
To reflect across the y-axis, we change the sign of the x-coordinates:
Reflected endpoints: (-1, 1) and (-4, 4)
So, the reflected endpoints are (-1, 1) and (-4, 4).
4. To reflect a point across the y-axis, we change the sign of its x-coordinate while keeping the y-coordinate the same.
The initial endpoints are (6, -3) and (3, 1).
To reflect across the y-axis, we change the sign of the x-coordinates:
Reflected endpoints: (-6, -3) and (-3, 1)
So, the new points after reflecting across the y-axis are (-6, -3) and (-3, 1).