If A Translation Of $T_{-3,-8}(x, Y)$ Is Applied To Square $A B C D$, What Is The $ Y Y Y [/tex]-coordinate Of $B$?A. $-12$B. $ − 8 -8 − 8 [/tex]C. $-6$D. $-2$

Introduction

In mathematics, geometric transformations play a crucial role in understanding the properties of shapes and their behavior under various transformations. One such transformation is the translation of a point or a shape by a given vector. In this article, we will explore the concept of translation and its application to a square $A B C D$. We will also determine the $y$-coordinate of point $B$ after applying the translation $T_{-3,-8}(x, y)$.

Translation of a Point or Shape

A translation is a transformation that moves a point or a shape by a given vector. The translation vector is denoted by $(h, k)$, where $h$ is the horizontal component and $k$ is the vertical component. When a point $(x, y)$ is translated by the vector $(h, k)$, the new coordinates of the point are given by $(x + h, y + k)$.

Translation of Square $A B C D$

Let's consider the square $A B C D$ with vertices at $(0, 0)$, $(3, 0)$, $(3, 3)$, and $(0, 3)$. We want to apply the translation $T_{-3,-8}(x, y)$ to this square. This means that we need to add $-3$ to the $x$-coordinate and $-8$ to the $y$-coordinate of each vertex.

Applying the Translation

To apply the translation, we need to add $-3$ to the $x$-coordinate and $-8$ to the $y$-coordinate of each vertex. The new coordinates of the vertices are:

• $A(0, 0) \rightarrow A'(-3, -8)$
• $B(3, 0) \rightarrow B'(-3 + 3, -8 + 0) = B'(-3, -8)$
• $C(3, 3) \rightarrow C'(-3 + 3, -8 + 3) = C'(-3, -5)$
• $D(0, 3) \rightarrow D'(-3 + 0, -8 + 3) = D'(-3, -5)$

Determining the $y$-Coordinate of $B$

We are asked to find the $y$-coordinate of point $B$ after applying the translation $T_{-3,-8}(x, y)$. From the previous section, we know that the new coordinates of point $B$ are $B'(-3, -8)$. Therefore, the $y$-coordinate of point $B$ is $-8$.

Conclusion

In this article, we explored the concept of translation and its application to a square $A B C D$. We determined the $y$-coordinate of point $B$ after applying the translation $T_{-3,-8}(x, y)$. The $y$-coordinate of point $B$ is $-8$.

Answer

The correct answer is:

• B. $-8$

Final Thoughts

Introduction

In our previous article, we explored the concept of translation and its application to a square $A B C D$. We determined the $y$-coordinate of point $B$ after applying the translation $T_{-3,-8}(x, y)$. In this article, we will answer some frequently asked questions related to geometric transformations and coordinate geometry.

Q&A

Q1: What is the difference between a translation and a rotation?

A1: A translation is a transformation that moves a point or a shape by a given vector, whereas a rotation is a transformation that rotates a point or a shape around a fixed point by a given angle.

Q2: How do you apply a translation to a point or a shape?

A2: To apply a translation to a point or a shape, you need to add the horizontal component of the translation vector to the $x$-coordinate and the vertical component of the translation vector to the $y$-coordinate of each vertex.

Q3: What is the effect of a translation on the coordinates of a point?

A3: A translation changes the coordinates of a point by adding the horizontal component of the translation vector to the $x$-coordinate and the vertical component of the translation vector to the $y$-coordinate.

Q4: Can a translation be represented as a matrix?

A4: Yes, a translation can be represented as a matrix. The matrix for a translation is given by:

$$\begin{bmatrix} 1 & 0 & h \\ 0 & 1 & k \\ 0 & 0 & 1 \end{bmatrix}$$

where $(h, k)$ is the translation vector.

Q5: How do you find the inverse of a translation matrix?

A5: To find the inverse of a translation matrix, you need to negate the translation vector and replace the original translation vector with the negated vector.

Q6: What is the effect of a translation on the distance between two points?

A6: A translation does not change the distance between two points. The distance between two points remains the same after a translation.

Q7: Can a translation be combined with other transformations?

A7: Yes, a translation can be combined with other transformations such as rotation, scaling, and reflection. The order of the transformations matters, and the translation should be applied last.

Q8: How do you apply a translation to a 3D object?

A8: To apply a translation to a 3D object, you need to add the horizontal component of the translation vector to the $x$-coordinate, the vertical component of the translation vector to the $y$-coordinate, and the depth component of the translation vector to the $z$-coordinate of each vertex.

Q9: What is the effect of a translation on the orientation of a 3D object?

A9: A translation does not change the orientation of a 3D object. The orientation of a 3D object remains the same after a translation.

Q10: Can a translation be used to solve problems in computer graphics?

A10: Yes, a translation can be used to solve problems in computer graphics. Translations are used to move objects in 2D and 3D space, and they are an essential part of computer graphics.

Conclusion

In this article, we answered some frequently asked questions related to geometric transformations and coordinate geometry. We hope that this article has provided a clear understanding of the concept of translation and its application to geometric shapes.