Translation of a Triangle Calculator
How to use the ‘Translation of a Triangle’ Calculator’?
Follow these steps to use the translation of a triangle calculator:
Step 1: Toggle and select the option of your choice as ‘Single Translation’ or ‘Composition Translation’.
Step 2: Enter the coordinates of the three vertices of a triangle into the respective input boxes.
Step 3: Enter the value or units for the translation and select the direction for both the x-axis and y-axis. In case of composition translation, the second translation units as well as the directions need to be entered.
Step 4: Click on the ‘Solve’ button to obtain the coordinates of the vertices of a triangle after translation. In case of composition translation, coordinates after the first as well as the second translation are 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 a triangle on the coordinate plane. With the use of sliders, it can be translated into various units along the x and y axes.
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 in which the size of a triangle or geometric shape remains the same, but only the position changes. A triangle or a line can be moved upward, downward, right, left, or in more than one direction along the x-axis and y-axis on the coordinate plane.
How do we find the Coordinates of the Vertices of a Translated Triangle?
Consider a triangle ABC. and the coordinates of its vertices are
\(A\left( x_{1}, y_{1} \right), B\left( x_{2}, y_{2}\right)\text{ }and\text{ }C\left( x_{3}, y_{3} \right)\).
So, the coordinates of the vertices of this triangle after translating ‘a’ units left or right along the x-axis and ‘b’ units up or down along the y – axis is stated as:
\(A’\left( x_{1} \pm a,\text{ }y_{1}\text{ }\pm \text{ }b\right), B’ \left( x_{2}\text{ }\pm a, \text{ }y_{2}\text{ } \pm b\right)\text{ }and\text{ }C’\left( x_{3} \pm \text{ }a,\text{ }y_3{} \pm b \right)\)
When referring to the composition translation of a triangle ABC where
\(A\left( x_{1}, y_{1} \right), B\left( x_{2}, y_{2} \right)\text{ }and\text{ }C\left( x_{3}, y_{3} \right),\) is done using two movements \(\left\langle \pm \text{ } a, \pm\text{ }b\right\rangle\) and \(\left\langle \pm \text{ } c, \pm\text{ }d\right\rangle\)
The translated triangle is \(A”\left( x_{1}\text{ }\pm \text{ }a\pm \text{ }c,\text{ }y_{1}\pm \text{ }b\text{ }\pm d\text{ } \right), B” \left( x_{2} \pm\text{ }a\pm\text{ }c,\text{ }y_{2}\pm \text{ }b\text{ }\pm d\right)\text{ }and\text{ }C”\left( x_{3}\text{ }\pm \text{ }a\pm\text{ }c,\text{ }y_{3} \pm b\pm d\right)\)
Solved Examples
Example 1: The coordinates of a triangle are (10, 2), (4, 1), and (3, 4). Translate it to 9 units left and 10 units down.
Solution:
Graph the triangle ABC and then move each vertex 9 units left and 10 units down.
The rule for the translation is:
\(A\left( x_{1},\text{ }y_{1} \right)\longrightarrow A’\left( x_{1}\text{ }\pm\text{ }a,\text{ }y_{1} \pm b \right)\)
Hence, A(10, 2) becomes A’(10 – 9, 2 – 10) = A’(1, -8)
B(4, 1) becomes B’(4 – 9, 1 – 10) = B’(-5, -9)
C(3, 4) becomes C’(3 – 9, 4 – 10) = C’(-6, -6)
Example 2: Translate a triangle 3 units right and 4 units up. The coordinates of vertices of the triangle ABC are A(-1, 2), B(2, 2) and C(0, 4).
Solution:
Graph the triangle ABC and then move each vertex 3 units right and 4 units up.
The rule for the translation is:
\(A\left( x_{1},\text{ }y_{1} \right)\to A’\left( x_{1}\pm a,\text{ }y_{1}\pm b \right) \)
Hence, A(-1, 2) becomes A’(-1 + 3, 2 + 4) = A’(2, 6)
B(2, 2) becomes B’(2 + 3, 2 + 4) = B’(5, 6)
C(0, 4) becomes C’(0 + 3, 4 + 4) = C’(3, 8)
Example 3: Translate a triangle 2 units right and 3 units up. The coordinates of the vertices of a triangle ABC are A(-2, -2), B(3, -2), and C(1, 3), and then translate the resulting image 4 units right and 5 units down.
Solution:
Step 1:
Graph the triangle ABC and then move each vertex 2 units right and 3 units up.
The rule for the translation is:
\(A\left( x_{1},\text{ }y_{1} \right)\to A’\left( x_{1}\pm a,\text{ }y_{1}\pm b \right) \)
Hence, A(-2, -2) becomes A’(-2 + 2, -2 + 3) = A’(0, 1)
B(3, -2) becomes B’(3 + 2, -2 + 3) = B’(5, 1)
C(1, 3) becomes C’(1 + 2, 3 + 3) = C’(3, 6)
Step 2:
Now move vertex A’, B’ and C’ 7 units right and 5 up.
Rule for the translation is:
\(A’\left( x_{1} \pm \text{ }a, \text{ }y_{1} \pm b\right)\to A”\left( x_{1}\pm \text{ }a\text{ }\pm \text{ }c,\text{ }y_{1}\pm \text{ }b\pm \text{ }d \right) \)
A’(0,1) becomes A”(0 + 7, 1 + 5) = A”(7, 6)
B’(5, 1) becomes B”(5 + 7, 1 + 5) = B”(12, 6)
C’(3, 6) becomes C”(3 + 7, 6 + 5) = C”(10, 11)
When a figure or shape is made to move in one or more directions right, left, up or down leads to its transformation. This specific form of transformation that only leads to movement but not rotation, reflection or change in the size of a shape is termed as ‘translation’.
When a triangle is translated on the coordinate plane, its location is changed by specified units in a specified direction. The translation has no effect on the length of the sides of a triangle. All three points will move by the same amount of units during the translation. If one point on a triangle is translated by 5 units to the left, then other points also will move 5 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 on the coordinate plane is included under the category of translation. However, any transformation involving turning, rotation, reflection, or change in size does not qualify as a translation. The translated shapes (called images) are the same size as the original shape, indicating that translated images are congruent.
- Sliding a carton from one corner of a room to another
- A child swinging on the monkey bars
- Moving and placing a glass from one cabinet to another