Dilation of a Quadrilateral Calculator
How to Use the ‘Dilation of a Quadrilateral’ Calculator’?
Follow these steps to use the ‘dilation of a quadrilateral’ calculator:
Step 1: Choose any one from the three functionalities, ‘dilating a quadrilateral on the coordinate plane’, ‘identify dilation’, or ‘finding the length’.
Step 2: When the functionality of ‘dilating a quadrilateral’ is selected, enter the coordinates of the vertices of a quadrilateral, and the value of the scale factor(k) into the respective input boxes. When the functionality ‘identify dilation’ is selected, enter the length of the sides of the object and image. When the functionality ‘finding the length’ is selected, enter any one of the side lengths of the object or image, and the value of the scale factor.
Step 3: Click on the ‘Solve’ button to obtain the required result.
Step 4: Click on the ‘Show Steps’ button to view the steps of the method applied to obtain the result.
Step 5: Click on the 
button to enter new inputs and start again.
Step 6: Click on the ‘Example’ button to play with random input values.
Step 7: Click on the ‘Explore’ button to drag and position the quadrilateral on the coordinate plane. Use the slider to dilate the quadrilateral for various scale factors.
Step 8: When on the ‘Explore’ page, click the ‘Calculate’ button to return to the calculator.
What Is Dilation?
A dilation is a type of transformation in which a geometric shape is either ‘enlarged’ or ‘reduced’ with respect to a fixed point C, known as the center of dilation. There is also the scale factor k, which is the ratio of the corresponding sides of the image to the length of its ‘preimage’.
What Is the Dilation of a Quadrilateral?
The dilation of a quadrilateral refers to the ‘relocation’ of a quadrilateral on the coordinate plane. This is done by calculating the horizontal and vertical distance of the vertices of a quadrilateral from the center of dilation, and then multiplying these distances with a scale factor to obtain the coordinates of the vertices of the dilated quadrilateral.
The dilation of a quadrilateral ABCD where the coordinates of the vertices are \(A\left( x_{1}, y_{1} \right), B\left( x_{2}, y_{2} \right), C\left( x_{3}, y_{3} \right)\text{ } and\text{ } D\left( x_{4}, y_{4} \right)\)with a scale factor k, is expressed as:
\(A\left( x_{1}, y_{1} \right) \to A’\left( kx_{1}, ky_{1} \right)\)
\(B\left( x_{2}, y_{2} \right) \to B’\left( kx_{2}, ky_{2} \right)\)
\(C\left( x_{3}, y_{3} \right) \to C’\left( kx_{3}, ky_{3} \right)\)
\(D\left( x_{4}, y_{4} \right) \to D’\left( kx_{4}, ky_{4} \right)\)
So, the coordinates of the vertices of a dilated quadrilateral A’B’C’D’ would be \(A’\left( kx_{1}, ky_{1} \right), B’\left( kx_{2}, ky_{2} \right), C’\left( kx_{3}, ky_{3} \right)\text{ } and\text{ } D’\left( kx_{4}, ky_{4} \right)\)
Note: The value of ‘k’ could be an integer, a decimal or a fraction.
Solved Examples
Example 1: The coordinates of the vertices of a quadrilateral are (1, 2), (6, 3), (5, 7) and (2, 6). Use scale factor as 2 and find the coordinates of the dilated quadrilateral.
Solution:
Let us assume that the vertices of the quadrilateral are A, B, C and D
So, A(1, 2), B (6, 3) , C(5, 7) and D(2, 6)
Applying the coordinate rule for dilation with scale factor, k = 2,
\(\left( x, y \right) \to \left( 2x, 2y \right)\)
\(A\left( 1, 2 \right) \to A’\left( 2, 4 \right) \)
\(B\left( 6, 3 \right)\to B’\left( 12, 6 \right)\)
\(C\left( 5, 7 \right)\to C’\left( 10, 14 \right)\)
\(D\left( 2, 6 \right)\to D’\left( 4, 12 \right)\)
So, the coordinates of vertices of the dilated quadrilateral A’B’C’D’ are A'(2, 4), B'(12, 6), C'(10, 14) and D'(4, 12).
Example 2: (-4, -4), (4, -4), (4, 4) and (-4, 4) are the vertices of a quadrilateral. Use a scale factor of 1.5 and find the coordinates of the dilated quadrilateral.
Solution:
Consider the vertices of the quadrilateral as A, B, C and D.
So, A(-4, -4), B (4, -4), C(4, 4) and D(-4, 4).
Applying the coordinate rule for dilation with scale factor, k = 1.5,
\(\left( x, y \right) \to \left( 1.5x, 1.5y \right)\)
\(A\left( -\text{ }4, -\text{ }4 \right)\to A’\left( -\text{ }6, -\text{ }6 \right)\)
\(B \left( 4, -\text{ }4 \right)\to B’\left( 6, -\text{ }6 \right)\)
\(C\left( 4, 4 \right)\to C’\left( 6, 6 \right)\)
\(D \left( -\text{ }4, 4 \right)\to D’ \left( -\text{ }6, 6 \right)\)
So, the coordinates of the vertices of the dilated quadrilateral A’B’C’D’ are A'(-6, -6), B'(6, -6), C'(6, 6) and D'(-6, 6).
Example 3: Calculate the scale factor and identify the dilation, if one of the sides of the object and the image of a quadrilateral are 6 yards and 9 yards.
Solution:
Object length = 6 yd
Image length = 9 yd
We know that,
\(k = \frac{Image\text{ }length}{Object\text{ }length}\)
Here, \(k = \frac{9}{6} = 1.5\)
As the scale factor is 1.5, the dilation is an ‘enlargement’.
Example 4: Determine the length of a rectangle if its image is 45 feet long after dilation. The scale factor used for dilation is 3.
Solution:
Scale factor, k = 3
We know that,
\(\frac{Image\text{ }length}{Object\text{ }length} = k\)
\(\frac{45}{Object\text{ }length} = 15\)
Object length=15
So, the rectangle is 15 feet long.
Example 5: The edge length of the image is 7 centimeters and that of the object is 28 centimeters. Calculate the scale factor and identify the dilation.
Solution:
Object length = 28 cm
Image length = 7 cm
We know that,
\(k = \frac{Image\text{ }length}{Object\text{ }length} \)
Here, \(k = \frac{7}{28} = 0.25 \)
As the scale factor is 0.25 so the dilation is a ‘reduction’.
The center of dilation is a point on the coordinate plane, about which a shape or figure is enlarged or reduced in size.
Enlarging or reducing the size of a geometric shape without altering its shape is known as dilation. Utilizing the scale factor provided for dilation aids in altering the size of a shape or object.
A scale factor is a number that defines the transformation of a shape or object. It is the ratio of the measure of the dimensions of an image to its preimage.
On the basis of the value of the scale factor(k), dilation is related to ‘enlargement’ or ‘reduction’ in the size of a shape.
For ‘k > 1’, dilation is an enlargement, and for ‘0 < k < 1’, the dilation is a reduction.
A dilation is a reduction or increase in the size of a shape or object. But translation is the movement of an object or figure along the direction of the ‘x’ and ‘y’ axes. In dilation, the size of the shape changes, but in translation, only the position of a shape changes on the coordinate plane but not the size.