Translation of the Line Calculator
How to use the ‘Translation of the Line’ Calculator’?
Follow the steps below to use the translation of the line calculator:
Step 1: Toggle and select the option of your choice as ‘Single Translation’ or ‘Composition Translation’.
Step 2: Enter the coordinates of the two points of the line into the respective input boxes.
Step 3: Enter the value or units for the translation and also, select the direction for both the x-axis and y-axis. In case of composition translation, enter the second translation units and select the direction as well.
Step 4: Click on the ‘Solve’ button to obtain the coordinates of the points of the line after translation. In case of composition translation, coordinates after the first as well as the second translation is calculated.
Step 5: Click on the ‘Show steps’ button to see the translation in a coordinate plane.
Step 6: Click on the refresh button to enter new inputs and start again.
Step 7: Click on the ‘Example’ button to play with different random input values.
Step 8: Click on the ‘Explore’ button to drag and position the line on the coordinate plane. With the use of sliders, it can be
translated for various units.
Step 9: When on the ‘Explore’ page, click on the ‘Calculate’ button if you want to go back to the calculator.
What is the Translation of a Line?
Translation is a specific form of transformation on the coordinate plane where size of the line or geometrical shape remains the same but only the position changes. The line or the line can be moved upward, downward, right, left or in more than one direction along the x-axis and y-axis in the coordinate plane.
How do we find the Coordinates of a Translated Line?
Consider a line AB and the coordinates of its endpoints are
\(A\left ( x_{1}, \text{ } y_{1} \right ) \)and \(B\left ( x_{2}, \text{ } y_{2} \right )\)
So, the coordinates of an end points of a line after translating a unit left or right along x-axis and b unit up or down along y – axis will be given as:
\(A’\left ( x_{1}\pm a, \text{ } y_{1}\pm b \right )\) and \(B’\left ( x_{2}\pm a, \text{ } y_{2}\pm b \right )\)
When composition translation of a line \(A\left ( x_{1},\text{ } y_{1} \right )\) and \(B\left ( x_{2},\text{ } y_{2} \right )\) is done using the two movements \(\left \langle \pm \text{ } a, \text{ } \pm \text{ } b \right \rangle\) and \(\left \langle \pm \text{ } c, \pm \text{ } d \right \rangle\)
The translated line is \(A” \left ( x_{1}\pm a\pm c,\text{ } y_{1}\pm b\pm d \right )\) and \(B” \left ( x_{2}\pm a\pm c,\text{ } y_{2}\pm b\pm d \right )\)
Solved Examples
Example 1: The coordinates of endpoints of a line is (1, 5)and (5, 1). Translate the line 9 units left and 1 unit down.
Solution:
Graph the line AB and then move each endpoint 9 units left and 1 unit down.
Rule for the translation is
\(A’ \left( x_{1}\text{ }\pm \text{ }a,\text{ }y_{1}\pm b \right)\) and \(B’ \left( x_{2}\text{ }\pm \text{ }a,\text{ }y_{2}\pm b \right)\)
Hence, \(A(1, \text{ } 5)\) becomes \(A'(1\text{ } -\text{ } 9, \text{ } 5\text{ } -\text{ } 1)= A’\left ( -8, \text{ } 4 \right )\)
And \(B(5, \text{ } 1)\) becomes \(B'(5\text{ } -\text{ } 9, 1\text{ } -\text{ } 1)= B’\left ( -4, \text{ } 0 \right )\)
Example 2: The coordinates of endpoints of a line is \(\left ( 2, -3 \right )\) and \(\left ( 6, -5 \right )\).
Translate the line 3 units right and 1 unit down.
Solution:
Graph the line AB and then move each endpoint 3 units right and 1 unit down.
Rule for the translation is
\(A’\left ( x_{1} \pm a, \text{ }y_{1}\pm b\right )\) and \(B’\left ( x_{2} \pm a,\text{ }y_{2}\pm b\right )\)
Hence, \(A \left ( 2, \text{ } -3 \right )\) becomes \(A’\left ( 2\text{ } +\text{ } 3, \text{ } -3-1 \right )=A’\left ( 5, \text{ } -4 \right )\)
And \(B \left ( -6, \text{ } 5 \right )\) becomes \(B’\left ( -6\text{ } +\text{ } 3,\text{ } 5-1 \right )=B’\left ( -3, \text{ } 4 \right )\)
Example 3: The coordinates of endpoints of a line is (-5, 8) and (4, 5). Translate the line 3 units left and 2 unit up and then translate again 4 unit right and 2 unit up.
Solution:
Graph the line AB and then move each endpoint 3 units left and 2 unit up.
Rule for the translation is
\(A’\left ( x_{1} \pm a, \text{ }y_{1}\pm b\right )\) and \(B’\left ( x_{2} \pm a,\text{ }y_{2}\pm b\right )\)
Hence, \(A \left ( -5,\text{ }8 \right )\) becomes \(A’ \left ( -5 -3, \text{ }8 + 2 \right ) = A’ \left ( -8, 10 \right )\)
And \(B \left ( 4,5 \right )\) becomes \(B’\left ( 4-3, \text{ } 5+2 \right )=B’\left ( 1, 7 \right )\)
Now move endpoints A’ and B’ 4 units right and 2 units up.
Then, \(A’\left ( -8, 10 \right )\) becomes \(A”\left ( -8+4, 10+2 \right )=A’\left ( -4, 12 \right )\)
And \(B’ \left ( 1,\text{ }7 \right )\) becomes \(B” \left ( 1\text{ } +\text{ } 4, \text{ } 7\text{ } +\text{ } 2 \right )= B’\left ( 5, \text{ }9 \right )\)
When a figure or shape is made to move in the direction(s) right, left, up or down leads to its transformation mathematically. This specific form of transformation which only leads to movement but not rotation, reflection or change of size is termed as translation.
When a line is translated in the coordinate plane, its location is changed by a specified units in a specified direction. The translation has no effect on the length. Both points will move by the same amount of units during translation. If one point on the line translated by 4 units to the left, then other points also will move 4 units to the left.
A translation that moves a shape left, right, up, or down or in more than one direction along the x-axis and y-axis in the coordinate plane falls in the category of translation. However, any transformation involving turn, rotation, reflection, or change in size does not qualify for translation. The translated shapes (called image) are the same size as the original shape, indicating that translated images are congruent.
- Sliding a cartoon box from one corner to another
- A child sliding on the slide swing
- Moving and keeping a glass from one cabinet to another