Finding the New Coordinates of Vertex C

Oct 13, 2025 by miodeya.com 83 views

Understanding the Problem

In geometry, a translation is a transformation that moves a point or a shape from one location to another without changing its size or orientation. Given the vertices of a triangle $$A(3, -2)$$, $$B(0, 0)$$, and $$C(-5, -4)$$, we are asked to find the new coordinates of vertex $$C$$ after applying the translation $$(x + 1, y + 2)$$.

Translation Formula

To translate a point $$(x, y)$$ by a vector $$(h, k)$$, we use the formula:

(x^{\prime}, y^{\prime}) = (x + h, y + k)

In this case, the translation vector is $$(1, 2)$$, so we will add $$1$$ to the x-coordinate and $$2$$ to the y-coordinate of each vertex.

Finding the New Coordinates of Vertex C

To find the new coordinates of vertex $$C$$, we apply the translation formula:

(x^{\prime}, y^{\prime}) = (-5 + 1, -4 + 2)

(x^{\prime}, y^{\prime}) = (-4, -2)

Therefore, the new coordinates of vertex $$C$$ are $$C^{\prime}(-4, -2)$$.

Comparison with the Given Options

The given options are:

A. $$C^{\prime}(4, 0)$$ B. $$C^{\prime}(-4, -2)$$ C. $$C^{\prime}(-5, -6)$$ D. $$C^{\prime}(3, -4)$$

Comparing the calculated coordinates $$C^{\prime}(-4, -2)$$ with the given options, we can see that the correct answer is:

B. $$C^{\prime}(-4, -2)$$

Conclusion

In this problem, we applied the translation formula to find the new coordinates of vertex $$C$$ after applying the translation $$(x + 1, y + 2)$$. We compared the calculated coordinates with the given options and found that the correct answer is $$C^{\prime}(-4, -2)$$.

Key Takeaways

Translation is a transformation that moves a point or a shape from one location to another without changing its size or orientation.
The translation formula is $$(x^{\prime}, y^{\prime}) = (x + h, y + k)$$.
To find the new coordinates of a vertex after translation, we apply the translation formula using the given translation vector.

Practice Problems

Find the new coordinates of vertex $$A$$ after applying the translation $$(x - 2, y + 3)$$.
Find the new coordinates of vertex $$B$$ after applying the translation $$(x + 4, y - 1)$$.
Find the new coordinates of vertex $$C$$ after applying the translation $$(x - 1, y + 2)$$.

Solutions

$(x^{\prime}, y^{\prime}) = (3 - 2, -2 + 3)$ $(x^{\prime}, y^{\prime}) = (1, 1)$
$(x^{\prime}, y^{\prime}) = (0 + 4, 0 - 1)$ $(x^{\prime}, y^{\prime}) = (4, -1)$
$(x^{\prime}, y^{\prime}) = (-5 - 1, -4 + 2)$ $(x^{\prime}, y^{\prime}) = (-6, -2)$$<br/>$

Frequently Asked Questions

Q: What is translation in geometry?

A: Translation is a transformation that moves a point or a shape from one location to another without changing its size or orientation.

Q: How do I apply the translation formula?

A: To apply the translation formula, you need to add the translation vector to the coordinates of the point. The formula is $$(x^{\prime}, y^{\prime}) = (x + h, y + k)$$, where $$(x, y)$$ is the original point and $$(h, k)$$ is the translation vector.

Q: What is the translation vector?

A: The translation vector is a pair of numbers that represents the amount of movement in the x and y directions. For example, if the translation vector is $$(1, 2)$$, it means that the point will be moved 1 unit to the right and 2 units up.

Q: How do I find the new coordinates of a vertex after translation?

A: To find the new coordinates of a vertex after translation, you need to apply the translation formula using the given translation vector. For example, if the vertex is $$(x, y)$$ and the translation vector is $$(h, k)$$, the new coordinates will be $$(x + h, y + k)$$.

Q: What if the translation vector is negative?

A: If the translation vector is negative, it means that the point will be moved in the opposite direction. For example, if the translation vector is $$(-1, -2)$$, it means that the point will be moved 1 unit to the left and 2 units down.

Q: Can I apply multiple translations to a point?

A: Yes, you can apply multiple translations to a point. To do this, you need to add the translation vectors together. For example, if the first translation vector is $$(1, 2)$$ and the second translation vector is $$(3, 4)$$, the total translation vector will be $$(1 + 3, 2 + 4) = (4, 6)$$.

Q: How do I check my answers?

A: To check your answers, you can use the following methods:

Graph the original point and the translated point to see if they are in the correct position.
Use the translation formula to calculate the new coordinates and compare them with your answer.
Check if the translated point has the same size and orientation as the original point.

Q: What are some common mistakes to avoid?

A: Some common mistakes to avoid when working with translations include:

Forgetting to add the translation vector to the coordinates of the point.
Using the wrong translation vector.
Not checking your answers.

Q: Can I use translations to solve real-world problems?

A: Yes, you can use translations to solve real-world problems. For example, you can use translations to:

Move a object from one location to another.
Change the size or orientation of an object.
Create a new shape or design.

Conclusion

In this article, we have covered the basics of translations in geometry, including the translation formula, how to apply it, and how to check your answers. We have also answered some frequently asked questions and provided some tips for avoiding common mistakes. With practice and experience, you will become more comfortable working with translations and be able to apply them to solve real-world problems.

Practice Problems

Find the new coordinates of vertex $$A$$ after applying the translation $$(x - 2, y + 3)$$.
Find the new coordinates of vertex $$B$$ after applying the translation $$(x + 4, y - 1)$$.
Find the new coordinates of vertex $$C$$ after applying the translation $$(x - 1, y + 2)$$.

Solutions

$(x^{\prime}, y^{\prime}) = (3 - 2, -2 + 3)$ $(x^{\prime}, y^{\prime}) = (1, 1)$
$(x^{\prime}, y^{\prime}) = (0 + 4, 0 - 1)$ $(x^{\prime}, y^{\prime}) = (4, -1)$
$(x^{\prime}, y^{\prime}) = (-5 - 1, -4 + 2)$ $(x^{\prime}, y^{\prime}) = (-6, -2)$