Question

triangle def is rotated 180° clockwise about the origin, translated 4 units to the right, and reflected over the x-axis. the image that results is δdef. grid image of coordinate plane with triangle def what are the coordinates of vertex d on the image? a. (-1, 8) b. (1, 4) c. (-6, 8) d. (9, -4)

Explanation:

Step1: Find original coordinates of D

From the graph, vertex \( D \) has coordinates \( (3, 4) \).

Step2: Rotate \( 180^\circ \) clockwise about origin

The rule for \( 180^\circ \) rotation about the origin is \( (x, y) \to (-x, -y) \).
Applying this to \( D(3, 4) \):
\( (3, 4) \to (-3, -4) \).

Step3: Translate 4 units to the right

The rule for translation 4 units right is \( (x, y) \to (x + 4, y) \).
Applying this to \( (-3, -4) \):
\( (-3 + 4, -4) = (1, -4) \).

Step4: Reflect over the x - axis

The rule for reflection over the \( x \) - axis is \( (x, y) \to (x, -y) \).
Applying this to \( (1, -4) \):
\( (1, -(-4))=(1, 4) \)? Wait, no, wait. Wait, after translation we had \( (1, -4) \). Reflecting over \( x \) - axis: \( (x,y)\to(x, -y) \), so \( (1, -4)\to(1, 4) \)? Wait, no, maybe I made a mistake. Wait, let's re - check the original coordinates. Wait, maybe the original \( D \) is \( (3,4) \)? Wait, no, looking at the grid, maybe the original \( D \) is \( (3,4) \)? Wait, no, maybe I misread the grid. Wait, let's start over.

Wait, maybe the original coordinates of \( D \) are \( (3, 4) \). Let's do the rotation again. \( 180^\circ \) rotation: \( (x,y)\to(-x,-y) \), so \( (3,4)\to(-3,-4) \). Then translate 4 units right: \( x=-3 + 4 = 1 \), \( y=-4 \), so \( (1,-4) \). Then reflect over \( x \) - axis: the rule is \( (x,y)\to(x, -y) \), so \( (1,-4)\to(1, 4) \)? But that's not one of the options. Wait, maybe the original \( D \) is \( (3, - 4) \)? Wait, no, the graph is in the upper half. Wait, maybe I messed up the reflection. Wait, the problem says "reflected over the \( x \) - axis". Wait, maybe the original \( D \) is \( (3,4) \). Wait, let's check the options. Option B is \( (1,4) \)? Wait, the options are A. \((-1,8)\), B. \((1,4)\), C. \((-6,8)\), D. \((2,-4)\) (wait, the last option is cut off, but let's re - evaluate.

Wait, maybe the original coordinates of \( D \) are \( (3,4) \). Let's re - do the transformations:

1. Rotation \( 180^\circ \) clockwise about origin: \( (x,y)\to(-x,-y) \), so \( (3,4)\to(-3,-4) \).

1. Translation 4 units to the right: \( x=-3 + 4=1 \), \( y = - 4 \), so the point is \( (1,-4) \).

1. Reflection over \( x \) - axis: \( (x,y)\to(x, - y) \), so \( (1,-4)\to(1,4) \). Which is option B.

Answer:

B. \((1, 4)\)