Question

use the transformation shown below to answer 1 - 2.

1. which is a true statement about the transformation?

a. triangle abc was translated 8 units left and 3 units up.
b. the pre - image is in quadrant ii.
c. the orientation of the figure was changed.
d. the orientation of the vertices was changed.

1. which rule best represents the transformation?

a. $(x,y)\to(-x,-y)$
b. $(x,y)\to(x - 8,y + 3)$
c. $(x,y)\to(x + 8,y - 3)$
d. $(x,y)\to(8x,-3y)$

Explanation:

1.

• Option A: Translation is a rigid transformation. If we assume coordinates (for example, if \(B\) is at \((-6,3)\) and \(B'\) is at \((0,0)\)), the translation is \(8\) units right and \(3\) units down, not \(8\) units left and \(3\) units up.
• Option B: Quadrant II has \(x<0,y > 0\). The pre - image (triangle \(ABC\)) is in quadrant II.
• Option C: Translation is a rigid transformation that preserves orientation (the shape and the order of vertices).
• Option D: Translation preserves the orientation of vertices.

• Let's assume a general point \((x,y)\) in the pre - image (triangle \(ABC\)) and its corresponding point \((x',y')\) in the image (triangle \(A'B'C'\)).
• If we consider the horizontal and vertical changes. For a point \(P(x,y)\) in \(ABC\) and \(P'(x',y')\) in \(A'B'C'\), we observe that \(x'=x + 8\) (right - ward shift) and \(y'=y-3\) (down - ward shift).
• Option A: \((x,y)\to(-x,-y)\) is a rotation about the origin by \(180^{\circ}\).
• Option B: \((x,y)\to(x - 8,y + 3)\) is a left - ward and up - ward translation.
• Option C: \((x,y)\to(x + 8,y-3)\) is a right - ward (since \(x\) value increases) and down - ward (since \(y\) value decreases) translation which matches the transformation.
• Option D: \((x,y)\to(8x,-3y)\) is a non - rigid transformation (a stretch and a reflection) which is not a translation.

Answer:

B

2.