Rectangle EFGH Is Translated According To The Rule $T_{-5,9}(x, Y$\]. If The Coordinates Of The Pre-image Of Point H Are $(-2,-3$\], What Are The Coordinates Of $H^{\prime}$?A. $(7,-8$\] B. $(-7,6$\] C.

Understanding the Translation Rule

In mathematics, a translation is a transformation that moves a figure from one location to another without changing its size or shape. The translation rule is given by the formula $T_{a,b}(x, y) = (x + a, y + b)$, where $(a, b)$ is the translation vector. In this problem, we are given the translation rule $T_{-5,9}(x, y)$, which means that the figure will be translated 5 units to the left and 9 units up.

Applying the Translation Rule

To find the coordinates of $H^{\prime}$, we need to apply the translation rule to the pre-image of point H. The pre-image of point H has coordinates $(-2,-3)$. We will substitute these values into the translation rule to find the coordinates of $H^{\prime}$.

Step 1: Substitute the Pre-Image Coordinates into the Translation Rule

The translation rule is given by $T_{-5,9}(x, y) = (x + a, y + b)$. We will substitute $x = -2$ and $y = -3$ into the rule.

x = -2
y = -3
a = -5
b = 9

x_prime = x + a
y_prime = y + b

Step 2: Calculate the Coordinates of H'

We will calculate the coordinates of $H^{\prime}$ by evaluating the expressions for $x^{\prime}$ and $y^{\prime}$.

x_prime = -2 + (-5)
y_prime = -3 + 9

Step 3: Simplify the Expressions

We will simplify the expressions for $x^{\prime}$ and $y^{\prime}$ to find the coordinates of $H^{\prime}$.

x_prime = -7
y_prime = 6

Conclusion

The coordinates of $H^{\prime}$ are $(-7, 6)$. Therefore, the correct answer is B.

Answer Key

A. $(7,-8)$ B. $(-7,6)$ C. $(7,6)$

Discussion

This problem requires the application of the translation rule to find the coordinates of $H^{\prime}$. The translation rule is a fundamental concept in geometry, and understanding how to apply it is essential for solving problems involving transformations. In this problem, we used the translation rule to find the coordinates of $H^{\prime}$, which is a critical step in solving problems involving translations.

Tips and Tricks

• Make sure to understand the translation rule and how to apply it.
• Use the correct values for the pre-image coordinates and the translation vector.
• Simplify the expressions for $x^{\prime}$ and $y^{\prime}$ to find the coordinates of $H^{\prime}$.

Related Topics

• Translations in geometry
• Coordinate geometry
• Transformations in mathematics

Practice Problems

• Find the coordinates of $H^{\prime}$ if the pre-image of point H has coordinates $(3, 4)$ and the translation rule is $T_{2, -3}(x, y)$.
• Find the coordinates of $H^{\prime}$ if the pre-image of point H has coordinates $(-5, 2)$ and the translation rule is $T_{-2, 4}(x, y)$.

Conclusion

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Q: What is the translation rule?

A: The translation rule is a formula that describes how to move a figure from one location to another without changing its size or shape. The translation rule is given by the formula $T_{a,b}(x, y) = (x + a, y + b)$, where $(a, b)$ is the translation vector.

Q: How do I apply the translation rule?

A: To apply the translation rule, you need to substitute the pre-image coordinates into the formula and simplify the expressions to find the coordinates of the translated figure.

Q: What is the pre-image of a point?

A: The pre-image of a point is the original point before it is translated. In other words, it is the point that is being translated.

Q: What is the translation vector?

A: The translation vector is a pair of numbers $(a, b)$ that describes how to move the figure. The first number $a$ represents the horizontal translation, and the second number $b$ represents the vertical translation.

Q: How do I find the coordinates of the translated figure?

A: To find the coordinates of the translated figure, you need to substitute the pre-image coordinates into the translation rule and simplify the expressions. The resulting coordinates will be the coordinates of the translated figure.

Q: What if the translation vector is negative?

A: If the translation vector is negative, it means that the figure will be translated in the opposite direction. For example, if the translation vector is $(-5, 9)$, the figure will be translated 5 units to the left and 9 units up.

Q: Can I use the translation rule to translate a figure in three dimensions?

A: Yes, you can use the translation rule to translate a figure in three dimensions. The translation rule in three dimensions is given by the formula $T_{a,b,c}(x, y, z) = (x + a, y + b, z + c)$, where $(a, b, c)$ is the translation vector.

Q: How do I apply the translation rule to a figure with multiple points?

A: To apply the translation rule to a figure with multiple points, you need to substitute the pre-image coordinates of each point into the translation rule and simplify the expressions to find the coordinates of the translated figure.

Q: Can I use the translation rule to translate a figure that is not a rectangle?

A: Yes, you can use the translation rule to translate any figure, not just rectangles. The translation rule is a general formula that can be applied to any figure.

Q: What are some common mistakes to avoid when applying the translation rule?

A: Some common mistakes to avoid when applying the translation rule include:

• Not substituting the pre-image coordinates into the formula
• Not simplifying the expressions to find the coordinates of the translated figure
• Using the wrong translation vector
• Not considering the direction of the translation vector

Conclusion

In this article, we discussed some common questions and answers related to the translation rule. We covered topics such as the translation rule, pre-image coordinates, translation vector, and how to apply the translation rule to find the coordinates of the translated figure. We also discussed some common mistakes to avoid when applying the translation rule. With practice and experience, you will become proficient in solving problems involving translations.