Finding precise answers for dilate points on Khan Academy helps learners verify their work and deepen understanding of geometric transformations. This guide explains how to enter correct responses and why each step matters in coordinate geometry.
Khan Academy provides structured exercises where students practice dilations centered at the origin or other points, and the expected answers rely on scale factor and fixed center coordinates. Using the correct notation ensures that automated checks and instructors can accurately assess performance.
| Concept | Key Rule | Example with Center (0,0) | Example with Center (h,k) |
|---|---|---|---|
| Dilation from origin | (x, y) → (k·x, k·y) | Scale 2: (3,4) → (6,8) | — |
| Dilation from arbitrary center | Translate, scale, translate back | — | (x,y) around (h,k) with factor k: (h + k·(x−h), k + k·(y−k)) |
| Negative scale factor | Reflection through center plus size change | Scale −1: (2,3) → (−2,−3) about origin | Involves 180° rotation around center |
| Fractional scale factor | Reduces distance from center | Scale 0.5: (8,6) → (4,3) about origin | Shrinks figure toward center |
Understanding Dilate Points Transformation Rules
Dilations resize a figure while preserving its shape, and the center point stays fixed. The scale factor determines whether the image grows or shrinks and whether it flips direction when the factor is negative.
On Khan Academy, you must apply the transformation to each vertex and then input the new coordinates in the required format. Paying attention to parentheses, signs, and order of coordinates prevents common mistakes.
Performing Dilations About the Origin
When the center of dilation is at the origin, the rule is straightforward: multiply both coordinates by the scale factor. This method is ideal for practicing the core idea without extra translation steps.
For example, dilating triangle vertices (2,1), (4,3), and (1,5) by a factor of 3 yields (6,3), (12,9), and (3,15). Entering each new vertex in order ensures the platform recognizes the entire shape correctly.
Handling Dilations Around a Nonzero Center
When the center is not at the origin, you translate the figure so that the center moves to the origin, apply the scale factor, and then translate back. This sequence guarantees that distances from the center are scaled accurately.
For a center (h,k) and scale factor k, the formula becomes (h + k·(x−h), k + k·(y−k)). Khan Academy exercises often use integer coordinates so that transformed points remain on the grid and are easier to verify.
Common Errors and How to Avoid Them
- Forgetting to apply the scale factor to both x and y coordinates.
- Mixing up the order of coordinates when entering answers.
- Misplacing signs, especially with negative scale factors or negative coordinates.
- Skipping the translation steps when the center is not at the origin.
Mastering Coordinate Dilations for Advanced Practice
Consistent practice with varied centers and scale factors builds intuition for how figures move and resize. Use hints, review incorrect steps, and retake similar problems until the process feels automatic.
- Memorize the dilation formula for both origin-centered and arbitrary center cases.
- Check each transformed vertex before submitting your final answer.
- Use graph paper or digital tools to visualize the original and dilated shapes.
- Track patterns in how coordinates change to speed up future problems.
FAQ
Reader questions
How do I enter dilated coordinates correctly on Khan Academy?
Type each new vertex in the required ordered pair format, using parentheses and a comma with no spaces, and separate multiple points exactly as the exercise specifies.
What should I do if my answer is marked incorrect but my math looks right?
Check for small typos such as swapped coordinates, missing negative signs, or extra spaces, and confirm that you used the correct center and scale factor.
Can I use a formula instead of plotting points for dilation?
Yes, applying the transformation formula directly is valid and often faster, especially when the center is at the origin, but verify with a quick sketch if unsure.
Does Khan Academy accept fractions or decimals in answer boxes?
Enter fractions in the required format, typically using the slash key, and avoid rounding decimals unless the exercise explicitly requests rounded answers.