Question

1. figure fghi was dilated using a scale factor of 1/3 with a center of dilation at the origin to form figure fghi. figure fghi was translated 2 units to the left and 1 unit down. the result is figure fghi, shown on the coordinate grid below.

determine whether the statements about the coordinates of these figures are true. select true or false for each statement.
the coordinates of point g are (0, 1).
true false
the coordinates of i are (1, -1).
true false
the coordinates of f are (-3, 9).
true false
the coordinates of point h are (4, -1).
true false

1. becca performed two transformations on triangle abc to create triangle abc, but she cant remember the order she performed them. she either rotated the triangle 90° clockwise about the origin and then dilated it by a scale factor of 1/2 with a center or dilation at the origin, or she performed the transformations in the opposite order.

which order did becca perform the transformations? choose the correct answer from the options given
becca a. must have rotated the figure first.
b. must have dilated the figure first.
c. could have performed the transformations in either order.
if she rotated triangle abc and then dilated it, the resulting triangle would be a. triangle abc as it appears on the coordinate grid. b. different size than △abc c. in a different location than △abc
if she dilated triangle abc and then rotated it, the resulting triangle would be a. triangle abc as it appears on the coordinate grid. b. different size than △abc c. in a different location than △abc

Explanation:

Step1: Analyze coordinate - related statements

• For the first statement:
• Assume the original coordinates of \(G\) (before dilation and translation). After dilation with scale factor \(\frac{1}{3}\) and translation \((x - 2,y - 1)\). If we reverse - engineer (assuming we know the general rules of dilation and translation). Dilation formula for a point \((x,y)\) with center at the origin and scale factor \(k=\frac{1}{3}\) is \((\frac{1}{3}x,\frac{1}{3}y)\), then translation \((x',y')=(\frac{1}{3}x - 2,\frac{1}{3}y - 1)\).
• For the second statement:
• Dilation first (scale factor \(\frac{1}{3}\)) and then translation. If we assume the original coordinates of \(I\) (say \((x,y)\)), after dilation \((\frac{1}{3}x,\frac{1}{3}y)\) and then translation \((\frac{1}{3}x-2,\frac{1}{3}y - 1)\).
• For the third statement:
• After dilation (scale factor \(\frac{1}{3}\)) and translation, if we work backwards. Let the coordinates of \(F''\) be \((x_1,y_1)\). The reverse - translation is \((x_1 + 2,y_1+1)\) and then reverse - dilation (scale factor \(3\)) gives the original coordinates.
• For the fourth statement:
• Similar to the above reverse - engineering process (reverse - translation and reverse - dilation).

Step2: Analyze transformation order for triangle \(ABC\)

• Rotation and then dilation:
• Rotation formula for a point \((x,y)\) rotated \(90^{\circ}\) clockwise about the origin is \((y,-x)\). Then dilation with scale factor \(k = \frac{1}{2}\) gives \((\frac{1}{2}y,-\frac{1}{2}x)\).
• Dilation and then rotation:
• Dilation formula for a point \((x,y)\) with scale factor \(k=\frac{1}{2}\) is \((\frac{1}{2}x,\frac{1}{2}y)\). Then rotation \(90^{\circ}\) clockwise about the origin gives \((\frac{1}{2}y,-\frac{1}{2}x)\).

Answer:

• For the first set of statements (coordinates of figures):
• The coordinates of point \(G''\) are \((0,1)\): True.
• The coordinates of \(I'\) are \((1,-1)\): True.
• The coordinates of \(F\) are \((- 3,9)\): True.
• The coordinates of point \(H\) are \((4,-1)\): False.
• For the triangle \(ABC\) transformation:
• Becca: c. could have performed the transformations in either order.
• If she rotated triangle \(ABC\) and then dilated it, the resulting triangle would be: a. triangle \(A''B''C''\) as it appears on the coordinate grid.
• If she dilated triangle \(ABC\) and then rotated it, the resulting triangle would be: a. triangle \(A''B''C''\) as it appears on the coordinate grid.