Question

5 △def≅△lmn. which sequence of transformations can be performed on △def to show this?
a reflect △def across the y - axis. then translate the image 6 units up.
b translate △def 6 units up. then rotate the image 90° counterclockwise around the origin.
c rotate △def 90° clockwise around the origin. then rotate the image 90° clockwise around the origin.
d translate △def 6 units to the left. then rotate the image 90° clockwise around the origin.
6 sarah draws figure efgh in the coordinate plane. she rotates figure efgh 90° clockwise around vertex e to form image efgh. draw the image efgh on the coordinate plane.
7 △pqr has vertices p(-4,4), q(-2,3), and r(-5,2). △pqr is rotated 90° clockwise around the origin. then the image is translated 5 units down and reflected across the y - axis. what are the coordinates of the vertices of △pqr? show your work

Explanation:

Step1: Analyze option A

• Reflection across \(y -\)axis:
• The rule for reflecting a point \((x,y)\) across the \(y -\)axis is \((x,y)\to(-x,y)\). For \(\triangle DEF\) with vertices (assume \(D(x_1,y_1)\), \(E(x_2,y_2)\), \(F(x_3,y_3)\)), after reflection, the vertices become \((-x_1,y_1)\), \((-x_2,y_2)\), \((-x_3,y_3)\).
• Translation 6 units up:
• The rule for translating a point \((x,y)\) 6 units up is \((x,y)\to(x,y + 6)\). After reflection and then translation, the final - image vertices will match the vertices of \(\triangle LMN\) (by observing the position of the triangles in the coordinate - plane).

Step2: Analyze option B

• Translation 6 units up:
• The rule is \((x,y)\to(x,y + 6)\).
• Rotation \(90^{\circ}\) counter - clockwise around the origin:
• The rule for rotating a point \((x,y)\) \(90^{\circ}\) counter - clockwise around the origin is \((x,y)\to(-y,x)\). This combination of transformations will not map \(\triangle DEF\) to \(\triangle LMN\) as the orientation and position (based on the coordinate - plane figure) will be incorrect.

Step3: Analyze option C

• First \(90^{\circ}\) clockwise rotation:
• The rule for rotating a point \((x,y)\) \(90^{\circ}\) clockwise around the origin is \((x,y)\to(y,-x)\).
• Second \(90^{\circ}\) clockwise rotation:
• Rotating the already rotated point \((y,-x)\) \(90^{\circ}\) clockwise around the origin gives \((-x,-y)\). This double - rotation will not map \(\triangle DEF\) to \(\triangle LMN\) as per the coordinate - plane figure.

Step4: Analyze option D

• Translation 6 units to the left:
• The rule is \((x,y)\to(x - 6,y)\).
• Rotation \(90^{\circ}\) clockwise around the origin:
• The rule is \((x,y)\to(y,-x)\). This combination of transformations will not map \(\triangle DEF\) to \(\triangle LMN\) as the orientation and position (based on the coordinate - plane figure) will be wrong.

Answer:

A. Reflect \(\triangle DEF\) across the \(y -\)axis. Then translate the image 6 units up.