Translation of a Quadrilateral Calculator
How to Use the ‘Translation of a Quadrilateral’ Calculator?
Follow these steps to use the translation of a quadrilateral calculator:
Step 1: Toggle and select the option of your choice with respect to either ‘Single Translation’ or ‘Composition Translation’.
Step 2: Enter the coordinates of all four vertices of a quadrilateral into the respective input boxes.
Step 3: Enter the value or units for the translation and select the direction for both the ‘x’ and ‘y’axes. In case of composition translation, the second translation units and the direction need to be entered.
Step 4: Click on the ‘Solve’ button to obtain the coordinates of the vertices of a quadrilateral 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 along with the steps of the solution.
Step 6: Click on the 
button to enter new inputs and start again.
Step 7: Click on the ‘Example’ button to view the result of translation with different or random input values.
Step 8: Click on the ‘Explore’ button to drag and position a quadrilateral on the coordinate plane. With the use of sliders, the quadrilateral can be translated into various units along the ‘x’ and ‘y’ axes.
Step 9: When on the ‘Explore’ page, click on the ‘Calculate’ button to go back to the calculator.
What Is the Translation of a Quadrilateral?
Translation is a specific form of transformation on the coordinate plane in which the size of a quadrilateral or any other geometric shape remains the same, but only the position changes. A quadrilateral, a line or any other flat shape can be moved upward, downward, right, left, or in more than one direction along the x-axis and the y-axis on the coordinate plane.
How Do We Find the Coordinates of the Vertices of a Translated Quadrilateral?
Consider a quadrilateral ABCD and the coordinates of its vertices are \(A\left( x_{1}, y_{1} \right), B\left( x_{2}, y_{2} \right)\text{ }C\left( x_{3},y_{3} \right)\text{ }and\text{ } D\left( x_{4}, y_{4} \right).\)
So, the coordinates of the vertices of this quadrilateral after translating ‘a’ units left or right along the x-axis and ‘b’ units up or down along the y-axis is stated as:
\(A’ \left( x_{1} \pm a, y_{1} \pm b \right), B’\left( x_{2} \pm a, y_{2} \pm b\right), C’\left( x_{3} \pm a, y_{3} \pm b \right)\text{ }and\text{ }D’\left( x_{4} \pm a, y_{4} \pm b\right) \)
When referring to composition translation of a quadrilateral ABCD, in which \(A\left( x_{1}, y_{1}\right), B\left( x_{2}, y_{2} \right), C\left( x_{3}, y_{3}\right)\text{ }and\text{ } D\left( x_{4}, y_{4} \right).\)
This is done using two movements \(\left\langle \pm \text{ }a, \pm \text{ }b \right\rangle and \text{ }\left\langle \pm \text{ }c, \pm \text{ }d \right\rangle \)
The translated quadrilateral is \(A”\left( x_{1} \pm a\pm c,\text{ }y_{1}\pm b\pm d \right),\)
\(B”\left( x_{2} \pm a\pm c,\text{ }y_{2}\pm b\pm d \right),\) \(C”\left( x_{3} \pm a\pm c,\text{ }y_{3}\pm b\pm d \right)\) and \(D”\left( x_{4} \pm a\pm c,\text{ }y_{4}\pm b\pm d \right)\)
Solved Examples
Example 1: The coordinates of a quadrilateral are (2, 3), (4, 2), (5, 4), and (3, 5). Translate it to 5 units right and 3 units up.
Solution:
Graph the quadrilateral ABCD and then move each vertex 9 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(2,3) becomes A’(2 + 5, 3 + 3) = A’(7, 6)
B(4, 2) becomes B’(4 + 5, 2 + 3) = B’(9, 5)
C(5, 4) becomes C’(5 + 5, 4 + 3) = C’(10, 7)
D(3, 5) becomes D’(3 + 5, 5 + 3) = D’(8, 8)
Example 2: Translate a quadrilateral 3 units left and 4 units up. The coordinates of the vertices of the quadrilateral ABCD are A(-3, -5), B(2, -6), C(1, -2) and D(-2, -1).
Solution:
Graph the quadrilateral ABCD and then move each vertex 3 units left 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(-3, -5) becomes A’(-3 -3, -5 + 4) = A’(-6, -1)
B(2, -6) becomes B’(2 – 3, – 6 + 4) = B’(-1, -2)
C(1, 2) becomes C’(1 – 3, 2 + 4) = C’(-2, 6)
D(-2, -1) becomes D’(-2 -3, -1 + 4) = D’(-5, 3)
Example 3: Translate a quadrilateral 5 units left and 6 units up. The coordinates of the vertices of a quadrilateral ABCD are A(4, -3), B(7, -2), C(6, 3), and D(3, 4). In the next step translate the resulting image 7 units right and 5 units up.
Solution:
Step 1:
Graph the quadrilateral ABCD and then move each vertex 5 units left and 6 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(4, -3) becomes A’(4 – 5, -3 + 6) = A’(-1, 3)
B(7, -2) becomes B’(7 – 5, -2 + 6) = B’(2, 4)
C(6, 3) becomes C’(6 – 5, 3 + 6) = C’(1, 9)
D(3, 4) becomes D’(3 – 5, 4 + 6) = D’(-2, 10)
Step 2:
Now move vertex A’, B’ and C’ 7 units right and 5 up.
Rule for the translation is:
\(A’\left( x_{1} \pm a,\text{ }y_{1}\pm b\right)\to A”\left( x_{1}\pm a\pm c,\text{ }y_{1} \pm b\pm d \right)\)
A’(-1, 3) becomes A”(-1 + 7, 3 + 5) = A”(6, 8)
B’(2, 4) becomes B”(2 + 7, 4 + 5) = B”(9, 9)
C’(1, 9) becomes C”(1 + 7, 9 + 5) = C”(8, 14)
D’(-2, 10) becomes D”(-2 + 7, 10 + 5) = D”(5, 15)
When a figure or shape is made to move in one or more directions right, left, up or down, it leads to its transformation. This specific form of transformation that only leads to movement but not rotation, reflection or a change in the size of a shape is termed as ‘translation’.
When a quadrilateral 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 the quadrilateral. All four vertices move by the same number of units during the translation. If one point on a quadrilateral is translated by 3 units to the right, then, the other points will also move 3 units to the right.
A translation that moves a shape left, right, up or down, or in more than one direction along the x-axis and the 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 monkey bars
- Moving and placing a glass from one cabinet to another