Dilation of a Line Calculator
How to Use the ‘Dilation of a Line’ Calculator’?
Follow these steps to use the ‘dilation of a line’ calculator:
Step 1: Enter the coordinates of the endpoints of a line and the scale factor(k) into the respective input boxes.
Step 2: Click on the ‘Solve’ button to obtain the coordinates of the line after dilation.
Step 3: Click on the ‘Show Steps’ button to view the steps of finding the coordinates after dilation, and the representation of dilation on the coordinate plane.
Step 4: Click on the 
button to enter new inputs and start again.
Step 5: Click on the ‘Example’ button to view the dilation of a line with random input values.
Step 6: Click on the ‘Explore’ button to drag and position the line on the coordinate plane. Use the slider to dilate the line for various scale factors.
Step 7: When on the ‘Explore’ page, click on the ‘Calculate’ button to go back to the calculator.
What Is Dilation?
A dilation is a type of transformation in which a figure 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 Line?
The dilation of a line refers to the relocation of a line on the coordinate plane. This is done by calculating the horizontal and vertical distance of the endpoints of a line from the center of dilation and then multiplying these distances with a scale factor to get the coordinates of the endpoints of the dilated line.
The dilation of a line AB where A(x, y) and B(p, q), with a scale factor k, is expressed as:
\(A\left( x,y \right)\to A’ \left( kx, ky \right)\)
\(B\left( p, q \right)\to B’ \left( kp, kq \right)\)
So, the coordinates of a dilated line A' B' would be A'(kx, ky) and B'(kp, kq).
Note: The value of k could be an integer, a decimal or a fraction.
Solved Examples
Example 1: The coordinates of the endpoints of a line are (3, 2) and (6, 1). Use a scale factor of 4 and find the coordinates of the dilated line.
Solution:
Let us assume the endpoints of the line as A and B.
A(3, 2) and B(6, 1)
Use the coordinate rule for dilation with scale factor, k = 4,
\(\left( x, y \right)\to \left( kx, ky \right)\)
\(A \left( 3, 2 \right) \to A’ \left( 12, 8 \right)\)
\(B \left( 6, 1 \right) \to B’ \left( 24, 4 \right)\)
So, the coordinates of endpoints of a dilated line A’ B’ are A’(12, 8) and B’(24, 4).
The following graph is a representation of the initial point and the dialeted point.
Example 2: (-3, -2) and (0, 4) are the endpoints of a line. Use a scale factor of 3 and find the coordinates of the dilated line.
Solution:
Consider the endpoints of the line as A and B.
A(-3, -2) and B(0, 4).
Use the coordinate rule for dilation with scale factor, k = 3,
\(\left( x, y \right)\to \left( kx, ky \right)\)
\(A\left( -3, -2 \right)\to A’\left( -9, -6 \right)\)
\(B\left( 0, 4 \right)\to B’\left( 0, 12 \right)\)
So, the coordinates of the endpoints of the dilated line A‘B‘ are A'(-9, -6) and B‘(0, 12).
The following graph is a representation of the initial point and the dialeted point.
Example 3: A line measuring 3.5 centimeters, measures 7 centimeters after dilation. Calculate the scale factor and identify the dilation.
Solution:
Initial value = 3.5 cm
Final value = 7 cm
We know that,
\( k = \frac{Final\text{ }Length}{Initial\text{ }Length}\)
Here, \(k = \frac{7}{3.5} = \frac{2}{1} = 2\)
As the scale factor is 2 so the dilation is an ‘Enlargement’.
Example 4: One side of a triangle is 5 inches and becomes 3 inches after dilation. Calculate the scale factor and identify the dilation.
Solution:
Initial value = 5 in
Final value = 3 in
We know that,
\(k = \frac{Final \text{ } Length}{Initial\text{ }Length}\)
Here, \(k = \frac{3}{5} = 0.6\)
As the scale factor is 0.6, and, 0.6 < 1, so the Dilation is a ‘Reduction’.
Example 5: Determine the length of the image of an insect 3 millimeters in size seen through the magnifying glass. A magnifying glass magnifies the image of an object by four times its original size.
Solution:
Scale factor, k = 4
We know that,
\(\frac{Image\text{ }Length}{Object\text{ }Length} = k\)
\(\frac{Image\text{ }Length}{3\text{ }} = 4\)
Image Length = 12
So, the image length through the magnifying glass is 12 millimeters.
Example 6: Determine the length of an object if its image is 12 feet long after dilation. The scale factor used for dilation is 0.3.
Solution:
Scale factor, k = 0.3
We know that,
\(\frac{Image\text{ }Length}{Object\text{ }Length} = k\)
\(\frac{12}{Object\text{ }Length} = 0.3\)
Object Length = 40
So, the object is 40 feet long.
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 changing 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.