Dilation of a Triangle Calculator
How to Use the ‘Dilation of a Triangle’ Calculator’?
Follow these steps to use the ‘dilation of a triangle’ calculator:
Step 1: Enter the coordinates of the vertices of a triangle and the value of the scale factor(k) into the respective input boxes.
Step 2: Click on the ‘Solve’ button to obtain the coordinates of the triangle 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 triangle with random input values.
Step 6: Click on the ‘Explore’ button to drag and position the triangle on the coordinate plane. Use the slider to dilate the triangle 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 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 Triangle?
The dilation of a triangle refers to the ‘relocation’ of a triangle on the coordinate plane. This is done by calculating the horizontal and vertical distance of the vertices of a triangle from the center of dilation, and then multiplying these distances with a scale factor to obtain the coordinates of the vertices of the dilated triangle.
The dilation of a triangle ABC where the coordinates of the vertices are, \(A\left( x_{1}, y_{1} \right), B\left( x_{2}, y_{2} \right)\text{ }and\text{ }C\left( x_{3}, y_{3} \right),\) with a scale factor k, is expressed as:
\(A\left( x_{1}, y_{1} \right) \to A’\left( kx_{1}, ky_{1} \right)\)
\(B\left( x_{2}, y_{2} \right) \to B’\left( kx_{2}, ky_{2} \right)\)
\(C\left( x_{3}, y_{3} \right) \to C’\left( kx_{3}, ky_{3} \right)\)
So, the coordinates of the vertices of a dilated triangle A’B’C’ would be \(A’\left( kx_{1}, ky_{1} \right), B’\left( kx_{2}, ky_{2}\right)\text{ }and\text{ } C’\left( kx_{3}, ky_{3} \right).\)
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 triangle are (-4, -1), (-1, -1) and (-3, 2). Use a scale factor of 3 and find the coordinates of the dilated triangle.
Solution:
Let us assume that the vertices of the triangle are A,B and C.
A(-4, -1), B (-1, -1) and C(-3, 2)
Applying the coordinate rule for dilation with scale factor, k = 3,
\(\left( x,y \right)\to \left( kx, ky \right)\)
\(A\left( -4, -1 \right) \to A’\left( -12, -3 \right)\)
\(B\left( -1, -1 \right) \to B’\left( -3, -3 \right)\)
\(C\left( -3, 2 \right) \to C’\left( -9, 6 \right)\)
So, the coordinates of vertices of the dilated triangle A’B’C’ are \(A’\left( – 12, – 3 \right), B’\left( -3, -3 \right)and C’\left( -9, 6 \right).\)
Example 2: (0, 1), (3, 0) and (3, 3) are the vertices of a triangle. Use a scale factor of -2 and find the coordinates of the dilated triangle.
Solution:
Consider the vertices of the triangle as A, B and C.
So, A(0, 1), B (3, 0) and C(3, 3).
Applying the coordinate rule for dilation with scale factor, k = -2,
\(\left( x, y \right) \to \left( kx, ky \right)\)
\(A\left( 0, 1 \right)\to A’\left( 0, -2 \right)\)
\(B\left( 3, 0 \right)\to B’\left( -6, 0 \right)\)
\(C\left( 3, 3 \right)\to C’\left( -6, -6 \right)\)
So, the coordinates of the vertices of the dilated triangle A’B’C’ are \(A’\left( 0, -2 \right), B’\left( -6, 0 \right)\text{ }and\text{ }C’\left( -6, -6 \right).\)
Example 3: (0, 4), (4, 0) and (4, 0) are the vertices of a triangle. Use a scale factor of 4 and find the coordinates of the dilated triangle.
Solution:
Consider the vertices of the triangle as A,B and C.
So, A(0, 4), B (4, 0) and C(4, 0).
Applying the coordinate rule for dilation with scale factor, k = 4,
\(\left( x, y \right)\to \left( kx, ky \right)\)
\(A\left( 0, 4 \right)\to A’\left( 0, 16 \right)\)
\(B\left( 4, 0 \right)\to B’\left( 16, 0 \right)\)
\(C\left( 0, -4 \right)\to C’\left( 0, -16 \right)\)
So, the coordinates of the vertices of the dilated triangle A’ B’ C’ are A'(0, 16), B'(16, 0) and C'(0, -16).
Example 4: One edge of a triangle measuring 7 feet now measures 10.5 feet after dilation. Calculate the scale factor and identify the dilation.
Solution:
Object length = 7 ft
Image length = 10.5 ft
We know that,
\(k = \frac{Image\text{ }length}{Object\text{ }length}\)
Here, \(k = \frac{10.5}{7} = 1.5\)
As the scale factor is 1.5, the dilation is an ‘Enlargement’.
Example 5: Determine the length of an object if its image is 75 millimeters long after dilation. The scale factor used for dilation is 5.
Solution:
Scale factor, k = 5
We know that,
\(\frac{Image\text{ }length}{Object\text{ }length} = k\)
\(\frac{75}{Object\text{ }length} = 5\)
Object length = 15 mm
So, the object is 15 millimeters long.
Example 6: The edge length of the image is 1.7 feet and that of the object is 5.1 feet. Calculate the scale factor and identify the dilation.
Solution:
Object length = 5.1 ft
Image length = 1.7 ft
We know that,
\(k = \frac{Image\text{ }length}{Object\text{ }length}\)
Here, \(k = \frac{1.7}{5.1} = 0.33\)
As the scale factor is 0.33 so the dilation is a ‘Reduction’.
The center of dilation is a point in a 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 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.