How to Dilate a Triangle with a Scale Factor (Formula & Examples)
The concept of dilation of a triangle calculator is crucial for students learning geometry and coordinate transformations. Dilation helps us change the size of a triangle while preserving its shape. It plays a key role in similar triangles, scaling for models, and many competitive exams in mathematics, including JEE and school boards.
Dilation of a Triangle: Core Concept Explained
In geometry, dilation is a transformation that produces an image that is the same shape as the original figure, but is a different size. When you use a triangle dilation calculator, you are mathematically "scaling" the triangle larger or smaller from a fixed point—known as the center of dilation. This transformation is defined by a scale factor:
- If the scale factor is greater than 1, the triangle becomes larger (enlargement).
- If between 0 and 1, it becomes smaller (reduction).
- If negative, the triangle is also reflected through the center of dilation.
This process is fundamental for understanding similar triangles, coordinate transformations, and geometric modeling. At Vedantu, we make these concepts accessible, with clear steps, tools, and examples to help you visualize and practice dilation.
Dilation Formula for a Triangle
The general formula to dilate a point (x, y) about a center (a, b) with scale factor k is:
Dilated point: (x', y') = (a + k(x - a), b + k(y - b))
Apply this formula to each vertex of the triangle to find its dilated image.
| Triangle Vertex | Original Coordinates | Dilated Coordinates |
|---|---|---|
| A | (x₁, y₁) | (a + k(x₁ - a), b + k(y₁ - b)) |
| B | (x₂, y₂) | (a + k(x₂ - a), b + k(y₂ - b)) |
| C | (x₃, y₃) | (a + k(x₃ - a), b + k(y₃ - b)) |
Where k is the scale factor and (a, b) is the chosen center of dilation (often the origin).
Worked Examples: Dilation Step-by-Step
Let's solve an example using the dilation of a triangle calculator:
Example 1: Dilation with Center at Origin
Given triangle ABC with points A(2, 3), B(4, 1), C(3, 5), dilate by scale factor 2 about the origin (0, 0).
-
Calculate new vertex A':
- x' = 0 + 2 × (2 - 0) = 4
- y' = 0 + 2 × (3 - 0) = 6
- So, A'(4, 6)
-
Calculate new vertex B':
- x' = 0 + 2 × (4 - 0) = 8
- y' = 0 + 2 × (1 - 0) = 2
- So, B'(8, 2)
-
Calculate new vertex C':
- x' = 0 + 2 × (3 - 0) = 6
- y' = 0 + 2 × (5 - 0) = 10
- So, C'(6, 10)
The dilated triangle A'(4,6), B'(8,2), C'(6,10) is twice as far from the origin as the original triangle.
Example 2: Dilation about a Point Other than Origin
Triangle DEF has D(1,2), E(4,2), F(2,5). Center of dilation is (1,2), scale factor 0.5 (shrink).
- D is at center, so D' = (1,2)
-
E':
x' = 1 + 0.5 × (4 - 1) = 1 + 0.5×3 = 2.5
y' = 2 + 0.5 × (2 - 2) = 2
E'(2.5, 2) -
F':
x' = 1 + 0.5 × (2 - 1) = 1.5
y' = 2 + 0.5 × (5 - 2) = 3.5
F'(1.5, 3.5)
You can check your answers or experiment with different scale factors quickly using online dilation calculators or the Vedantu practice app.
Practice Problems
- Dilate triangle XYZ with vertices (0,0), (3,4), (4,1) by a scale factor of 3 about the origin.
- Dilate triangle PQR with vertices (2,2), (5,2), (3,6) using a center of dilation (2,2) and a scale factor of 0.5.
- What happens to triangle JKL (J(1,1), K(3,1), L(2,4)) if dilated with scale factor -1 about the origin?
- Use a triangle dilation calculator to check your solution to problem (1).
- Dilate the point M(7, -5) about (4, -2) with a scale factor of 2.
Common Mistakes to Avoid
- Forgetting to subtract the center coordinates when using a non-origin center.
- Confusing a negative scale factor with a positive (negative reflects the shape).
- Applying the scale factor to only x OR y, instead of both.
- Not checking if angles have changed (they should not—angles are preserved under dilation!)
Real-World Applications
Triangle dilation is used in graphic design, architecture, map making (scale models), and in mathematical proofs involving similar triangles. It also appears in art enlargements and model building. At Vedantu, our online tools and lessons help students apply these concepts in real-world and exam scenarios.
Explore more about scale factor and transformations to understand how shapes and figures change under different operations.
In this page, you learned the essentials of the dilation of a triangle calculator—the formula, step-by-step working, practice problems, and real-life value. Dilation is a key concept for geometry success and similar triangles. At Vedantu, we're here to help you master every transformation, whether for school, exams, or deeper understanding.
FAQs on Triangle Dilation Calculator & Step-by-Step Guide
1. What is the formula for the dilation of a triangle?
The dilation formula for a triangle involves multiplying the coordinates of each vertex by a scale factor (k) relative to a center of dilation. For a point (x, y) and center (a, b), the dilated point (x', y') is calculated as: (x', y') = (a + k(x - a), b + k(y - b)). This formula ensures accurate scaling and transformation of triangles.
2. How to dilate a triangle by 3?
To dilate a triangle by a scale factor of 3, apply the dilation formula to each vertex. Identify the center of dilation. For each vertex (x, y), calculate the new coordinates (x', y') using the formula: (x', y') = (center_x + 3(x - center_x), center_y + 3(y - center_y)). Connect the new vertices to form the dilated triangle. This process enlarges the triangle while maintaining its shape and angles.
3. How to dilate a triangle by 2?
Dilating a triangle by a scale factor of 2 is similar to dilating by 3. You'll use the dilation formula, replacing 'k' with 2. Specifically, for each vertex (x, y) and center of dilation (a, b), the new coordinates (x', y') will be: (x', y') = (a + 2(x - a), b + 2(y - b)). This doubles the distance of each vertex from the center of dilation, resulting in a larger, similar triangle.
4. How to find the 3rd side of a triangle?
Finding the third side depends on the information you have. If you know two sides (a, b) and the angle (C) between them, use the Law of Cosines: c² = a² + b² - 2ab cos(C). If you know two sides and one angle opposite a known side (SSA), there might be 0, 1, or 2 solutions using the Law of Sines. With three sides (SSS), you can use the Law of Cosines to find angles first and then any side.
5. What is the formula for the dilation of a triangle?
The dilation formula is used to find the new coordinates of a triangle's vertices after dilation. Given a point (x, y), a center of dilation (a, b), and a scale factor (k), the new coordinates (x', y') are: (x', y') = (a + k(x - a), b + k(y - b)). This formula applies to each vertex of the triangle.
6. How do I dilate a triangle using the calculator?
Using a dilation calculator simplifies the process. Input the coordinates of each vertex of your triangle, specify the coordinates of the center of dilation, and enter the scale factor. The calculator will compute the new coordinates of the dilated triangle. This is a quick way to check your work or to solve complex problems.
7. What is dilation of a triangle?
Dilation is a transformation that enlarges or shrinks a geometric figure. For a triangle, dilation involves multiplying the distance of each vertex from a fixed point (the center of dilation) by a constant scale factor (k). If k > 1, the triangle is enlarged; if 0 < k < 1, it's shrunk; and if k < 0, it is enlarged and reflected.
8. What happens if the scale factor is negative?
A negative scale factor (k) in a triangle dilation reflects the image across the center of dilation while also scaling it. The size changes based on the magnitude of k ( |k| ), but the orientation of the triangle is reversed. For example, if k = -2, the image will be twice as large but flipped.
9. How does dilation affect the angles of a triangle?
Dilation preserves the angles of a triangle. The dilated triangle is similar to the original, meaning that corresponding angles are congruent. Only the lengths of the sides change proportionally, according to the scale factor. This is a crucial property of dilation in geometry.
10. Can you dilate a triangle about any point?
Yes, the center of dilation can be any point—inside, outside, or on the triangle itself. The choice of center affects the position of the dilated triangle, but the scale factor determines its size relative to the original. The formula remains consistent regardless of the center's location.
