Question

topic 3 assessment: transformations

1. what is a rule for the translation of △rst?

• (x,y)→(x + 7,y - 3)
• (x,y)→(x + 3,y - 7)
• (x,y)→(x - 3,y + 7)
• (x,y)→(x - 7,y + 3)

1. what rigid motion maps the solid - line figure onto the dotted - line figure? circle one

(translation, rotation, reflection, dilation) (translation, rotation, reflection, dilation)

Explanation:

Step1: Analyze the translation of $\triangle RST$

Let's take a point, say \(R\). Suppose the original coordinates of \(R\) (before translation) and \(R'\) (after translation). If we assume \(R=(x_1,y_1)\) and \(R'=(x_2,y_2)\). Count the horizontal and vertical shifts.
The horizontal shift: If we move from the left - hand side \(R\) to the right - hand side \(R'\), we move \(7\) units to the right (since \(x\) - coordinate increases). The vertical shift: We move \(3\) units down (since \(y\) - coordinate decreases).
For a general point \((x,y)\), moving \(a\) units to the right (if \(a>0\)) in the \(x\) - direction and \(b\) units down (if \(b > 0\)) in the \(y\) - direction, the rule is \((x,y)\to(x + a,y - b)\). Here \(a = 7\) and \(b=3\), so the rule is \((x,y)\to(x + 7,y - 3)\)

Step2: Analyze the rigid motion for the first figure (lightning - shaped)

• Translation: A translation is a slide. But the orientation of the solid - line and dotted - line lightning - shaped figures is different.
• Rotation: A rotation changes the orientation of a figure. If we rotate the solid - line lightning - shaped figure, we can map it onto the dotted - line figure.
• Reflection: A reflection would flip the figure over a line, which is not the case here as the orientation change is more of a turn than a flip.
• Dilation: A dilation changes the size of a figure. Since the size of the two lightning - shaped figures is the same, dilation is not the answer.

Step3: Analyze the rigid motion for the second figure (tree - shaped)

• Translation: A translation is a slide. The orientation of the solid - line and dotted - line tree - shaped figures is the same, but the position is different. A translation (a slide) can map the dotted - line figure onto the solid - line figure.
• Rotation: There is no turn (rotation) needed as the orientation is the same.
• Reflection: A reflection would flip the figure over a line, which is not the case here as there is no flip.
• Dilation: A dilation changes the size of a figure. Since the size of the two tree - shaped figures is the same, dilation is not the answer.

Answer:

1. \((x,y)\to(x + 7,y - 3)\)
2. Rotation, Translation