Question

1. select all of the transformations that would carry fghj onto itself. (image below)

reflection over the y-axis
reflection over the line x=-1
reflection over the line y=-2
rotation of 90° clockwise about the origin
rotation of 180° counterclockwise about the point (-1,-4)
rotation of 360° counterclockwise about point g
image of a grid with f, g, h, j

1. triangle qrs is a rigid motion transformation of triangle jkl. the side lengths, in centimeters (cm), of triangle jkl are given below. hint* draw the triangles.

jk = 7 cm
kl = 12 cm
lj = 8 cm
what are the lengths, in centimeters, of each side of triangle qrs?
qr = ______
rs = ______
sq = ______

Explanation:

Question 2 (Transformations carrying FGHJ onto itself)

To determine which transformations carry the figure FGHJ onto itself, we analyze each option:

1. Reflection over the y - axis: The figure FGHJ is not symmetric about the y - axis. So this reflection will not carry it onto itself.
2. Reflection over the line \(x = - 1\): The line \(x=-1\) is the vertical line of symmetry for the rectangle FGHJ (from the grid, we can see that the figure is symmetric about \(x = - 1\)). So a reflection over \(x=-1\) will map the figure onto itself.
3. Reflection over the line \(y=-2\): The line \(y = - 2\) is the horizontal line of symmetry for the rectangle FGHJ (from the grid, the figure is symmetric about \(y=-2\)). So a reflection over \(y = - 2\) will map the figure onto itself.
4. Rotation of \(90^{\circ}\) clockwise about the origin: A \(90^{\circ}\) clockwise rotation about the origin will change the position of the rectangle in a way that it does not map onto itself.
5. Rotation of \(180^{\circ}\) counterclockwise about the point \((-1,-4)\): A \(180^{\circ}\) rotation about the center of the rectangle (which seems to be \((-1,-4)\) from the grid) will map the rectangle onto itself, as a \(180^{\circ}\) rotation about the center of a rectangle maps it onto itself.
6. Rotation of \(360^{\circ}\) counterclockwise about point G: A \(360^{\circ}\) rotation about any point maps the figure onto itself, since a \(360^{\circ}\) rotation is equivalent to no rotation.

So the transformations that carry FGHJ onto itself are: Reflection over the line \(x=-1\), Reflection over the line \(y = - 2\), Rotation of \(180^{\circ}\) counterclockwise about the point \((-1,-4)\), Rotation of \(360^{\circ}\) counterclockwise about point G.

Question 3 (Rigid motion and side lengths)

Rigid motion transformations (such as translation, rotation, reflection) preserve the side lengths of a figure. This means that if triangle QRS is a rigid motion transformation of triangle JKL, then the corresponding sides of the two triangles are equal in length.

• Corresponding to \(JK = 7\space cm\), the side \(QR\) (since \(JK\) and \(QR\) are corresponding sides in the rigid - motion transformation) will have length \(QR=JK = 7\space cm\).
• Corresponding to \(KL = 12\space cm\), the side \(RS\) (since \(KL\) and \(RS\) are corresponding sides) will have length \(RS = KL=12\space cm\).
• Corresponding to \(LJ = 8\space cm\), the side \(SQ\) (since \(LJ\) and \(SQ\) are corresponding sides) will have length \(SQ=LJ = 8\space cm\).

Final Answers

Question 2

The transformations are: Reflection over the line \(x=-1\), Reflection over the line \(y=-2\), Rotation of \(180^{\circ}\) counterclockwise about the point \((-1,-4)\), Rotation of \(360^{\circ}\) counterclockwise about point G.

Question 3

\(QR=\boldsymbol{7\space cm}\), \(RS=\boldsymbol{12\space cm}\), \(SQ=\boldsymbol{8\space cm}\)

Answer:

Question 2 (Transformations carrying FGHJ onto itself)

To determine which transformations carry the figure FGHJ onto itself, we analyze each option:

1. Reflection over the y - axis: The figure FGHJ is not symmetric about the y - axis. So this reflection will not carry it onto itself.
2. Reflection over the line \(x = - 1\): The line \(x=-1\) is the vertical line of symmetry for the rectangle FGHJ (from the grid, we can see that the figure is symmetric about \(x = - 1\)). So a reflection over \(x=-1\) will map the figure onto itself.
3. Reflection over the line \(y=-2\): The line \(y = - 2\) is the horizontal line of symmetry for the rectangle FGHJ (from the grid, the figure is symmetric about \(y=-2\)). So a reflection over \(y = - 2\) will map the figure onto itself.
4. Rotation of \(90^{\circ}\) clockwise about the origin: A \(90^{\circ}\) clockwise rotation about the origin will change the position of the rectangle in a way that it does not map onto itself.
5. Rotation of \(180^{\circ}\) counterclockwise about the point \((-1,-4)\): A \(180^{\circ}\) rotation about the center of the rectangle (which seems to be \((-1,-4)\) from the grid) will map the rectangle onto itself, as a \(180^{\circ}\) rotation about the center of a rectangle maps it onto itself.
6. Rotation of \(360^{\circ}\) counterclockwise about point G: A \(360^{\circ}\) rotation about any point maps the figure onto itself, since a \(360^{\circ}\) rotation is equivalent to no rotation.

So the transformations that carry FGHJ onto itself are: Reflection over the line \(x=-1\), Reflection over the line \(y = - 2\), Rotation of \(180^{\circ}\) counterclockwise about the point \((-1,-4)\), Rotation of \(360^{\circ}\) counterclockwise about point G.

Question 3 (Rigid motion and side lengths)

Rigid motion transformations (such as translation, rotation, reflection) preserve the side lengths of a figure. This means that if triangle QRS is a rigid motion transformation of triangle JKL, then the corresponding sides of the two triangles are equal in length.

• Corresponding to \(JK = 7\space cm\), the side \(QR\) (since \(JK\) and \(QR\) are corresponding sides in the rigid - motion transformation) will have length \(QR=JK = 7\space cm\).
• Corresponding to \(KL = 12\space cm\), the side \(RS\) (since \(KL\) and \(RS\) are corresponding sides) will have length \(RS = KL=12\space cm\).
• Corresponding to \(LJ = 8\space cm\), the side \(SQ\) (since \(LJ\) and \(SQ\) are corresponding sides) will have length \(SQ=LJ = 8\space cm\).

Final Answers

Question 2

The transformations are: Reflection over the line \(x=-1\), Reflection over the line \(y=-2\), Rotation of \(180^{\circ}\) counterclockwise about the point \((-1,-4)\), Rotation of \(360^{\circ}\) counterclockwise about point G.

Question 3

\(QR=\boldsymbol{7\space cm}\), \(RS=\boldsymbol{12\space cm}\), \(SQ=\boldsymbol{8\space cm}\)