Hexagon DEFGHI Is Translated 8 Units Down And 3 Units To The Right. If The Coordinates Of The Pre-image Of Point $F$ Are $(-9, 2$\], What Are The Coordinates Of $F^{\prime}$?A. $(-17, 5$\] B. $(-6, -6$\] C.

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Introduction

In geometry, transformations are used to describe changes in the position of shapes and figures. One common type of transformation is a translation, which involves moving a point or shape from one location to another without changing its size or orientation. In this article, we will explore how to translate points in a coordinate plane, using the example of translating the hexagon DEFGHI 8 units down and 3 units to the right.

Understanding Coordinate Translations

To translate a point in a coordinate plane, we need to understand how to change its coordinates. The general formula for translating a point (x, y) by a units horizontally and b units vertically is:

(x', y') = (x + a, y + b)

where (x', y') are the new coordinates of the translated point.

Applying the Translation to the Hexagon DEFGHI

In this example, we are given the coordinates of the pre-image of point F as (-9, 2). We need to translate this point 8 units down and 3 units to the right. Using the formula above, we can calculate the new coordinates of F' as follows:

x' = x + a = -9 + 3 = -6 y' = y + b = 2 - 8 = -6

Therefore, the coordinates of F' are (-6, -6).

Conclusion

In this article, we have explored how to translate points in a coordinate plane using the example of translating the hexagon DEFGHI 8 units down and 3 units to the right. We have used the general formula for translating a point to calculate the new coordinates of F'. The coordinates of F' are (-6, -6).

Example Problems

  1. Translate the point (4, 5) 2 units left and 3 units up.
  2. Translate the point (-2, 1) 4 units right and 2 units down.
  3. Translate the point (0, 0) 5 units up and 3 units left.

Solutions

  1. The new coordinates of the translated point are (4 - 2, 5 + 3) = (2, 8).
  2. The new coordinates of the translated point are (-2 + 4, 1 - 2) = (2, -1).
  3. The new coordinates of the translated point are (0, 0 + 5) = (0, 5).

Practice Problems

  1. Translate the point (3, -2) 2 units right and 4 units up.
  2. Translate the point (-5, 3) 3 units left and 2 units down.
  3. Translate the point (1, 0) 4 units up and 2 units right.

Answers

  1. The new coordinates of the translated point are (3 + 2, -2 + 4) = (5, 2).
  2. The new coordinates of the translated point are (-5 - 3, 3 - 2) = (-8, 1).
  3. The new coordinates of the translated point are (1, 0 + 4) = (1, 4).

Tips and Tricks

  • When translating a point, make sure to add the horizontal and vertical components separately.
  • Use the general formula for translating a point to calculate the new coordinates.
  • Practice translating points in different directions and with different values to become more comfortable with the concept.

Common Mistakes

  • Forgetting to add the horizontal and vertical components separately.
  • Using the wrong formula or values.
  • Not checking the units of the translated point.

Conclusion

Q&A: Translating Points in a Coordinate Plane

Q: What is a translation in geometry?

A: A translation is a type of transformation that involves moving a point or shape from one location to another without changing its size or orientation.

Q: How do I translate a point in a coordinate plane?

A: To translate a point (x, y) by a units horizontally and b units vertically, you can use the formula:

(x', y') = (x + a, y + b)

where (x', y') are the new coordinates of the translated point.

Q: What is the difference between a translation and a rotation?

A: A translation involves moving a point or shape from one location to another without changing its size or orientation, whereas a rotation involves rotating a point or shape around a fixed point without changing its size or position.

Q: Can I translate a point by a negative value?

A: Yes, you can translate a point by a negative value. For example, if you want to translate a point 2 units to the left, you would use the formula:

(x', y') = (x - 2, y)

Q: How do I translate a point by a fraction of a unit?

A: To translate a point by a fraction of a unit, you can multiply the fraction by the unit value. For example, if you want to translate a point 1/2 unit up, you would use the formula:

(x', y') = (x, y + 1/2)

Q: Can I translate a point by a decimal value?

A: Yes, you can translate a point by a decimal value. For example, if you want to translate a point 3.5 units to the right, you would use the formula:

(x', y') = (x + 3.5, y)

Q: How do I translate a point by a negative decimal value?

A: To translate a point by a negative decimal value, you can simply add the negative value to the original coordinates. For example, if you want to translate a point -3.5 units to the left, you would use the formula:

(x', y') = (x - 3.5, y)

Q: Can I translate a point by a mixed number?

A: Yes, you can translate a point by a mixed number. For example, if you want to translate a point 2 1/2 units up, you would use the formula:

(x', y') = (x, y + 2 1/2)

Q: How do I translate a point by a fraction of a unit in a different direction?

A: To translate a point by a fraction of a unit in a different direction, you can use the formula:

(x', y') = (x + a, y + b)

where a and b are the horizontal and vertical components of the translation, respectively.

Q: Can I translate a point by a combination of translations?

A: Yes, you can translate a point by a combination of translations. For example, if you want to translate a point 2 units to the right and 3 units up, you would use the formula:

(x', y') = (x + 2, y + 3)

Q: How do I translate a point by a negative combination of translations?

A: To translate a point by a negative combination of translations, you can simply add the negative values to the original coordinates. For example, if you want to translate a point -2 units to the left and -3 units down, you would use the formula:

(x', y') = (x - 2, y - 3)

Conclusion

In conclusion, translating points in a coordinate plane is an essential concept in geometry. By understanding how to change the coordinates of a point, we can apply transformations to shapes and figures. In this article, we have provided answers to common questions about translating points in a coordinate plane, including how to translate a point by a unit value, a fraction of a unit, a decimal value, and a combination of translations.