Question

ing transformations
japal drew a triangle that had vertices with coordinates (1, -2), (4, -2), and (1, -4). he reflected the figure across the x - axis. what are the coordinates of the vertices of the new figure?
(-1, -2), (-4, -2), and (-1, -4)
(2, -1), (2, -4), and (4, -1)
(1, 2), (4, 2), and (1, 4)
(-1, 2), (-4, 2), and (-1, 4)

Explanation:

Step1: Recall reflection over x - axis rule

The rule for reflecting a point \((x,y)\) across the \(x\) - axis is \((x,y)\to(x, - y)\). Wait, no, wait. Wait, the original vertices are \((1,-2)\), \((4,-2)\), and \((1,-4)\)? Wait, no, the problem says Jaipal drew a triangle with vertices \((1, - 2)\), \((4,-2)\), and \((1,-4)\)? Wait, no, the user's image - text: "Jaipal drew a triangle that had vertices with coordinates \((1, - 2)\), \((4, - 2)\), and \((1, - 4)\). He reflected the figure across the \(x\) - axis. What are the coordinates of the vertices of the new figure?" Wait, no, maybe I misread. Wait, the reflection over \(x\) - axis: the formula is \((x,y)\to(x, - y)\). Wait, original points: \((1,-2)\), \((4,-2)\), \((1,-4)\). Let's apply the reflection over \(x\) - axis. For a point \((x,y)\), after reflecting over \(x\) - axis, the \(x\) - coordinate remains the same, and the \(y\) - coordinate is multiplied by \(- 1\). So:

For \((1,-2)\): \(x = 1\), \(y=-2\), new \(y=-(-2)=2\), so the point becomes \((1,2)\).

For \((4,-2)\): \(x = 4\), \(y = - 2\), new \(y=-(-2)=2\), so the point becomes \((4,2)\).

For \((1,-4)\): \(x = 1\), \(y=-4\), new \(y=-(-4)=4\), so the point becomes \((1,4)\).

Looking at the options, the third option is \((1,2)\), \((4,2)\), and \((1,4)\).

Step2: Verify each option

• First option: \((-1,-2)\), \((-4,-2)\), \((-1,-4)\): These are reflections over \(y\) - axis (since \(x\) - coordinates are negated), not \(x\) - axis. So incorrect.
• Second option: \((2,-1)\), \((2,-4)\), \((4,-1)\): These do not follow the reflection over \(x\) - axis rule. Incorrect.
• Third option: \((1,2)\), \((4,2)\), \((1,4)\): Follows the reflection over \(x\) - axis rule (\(y\) - coordinates are negated). Correct.
• Fourth option: \((-1,2)\), \((-4,2)\), \((-1,4)\): These are reflections over both \(x\) and \(y\) axes ( \(x\) negated and \(y\) negated), not just \(x\) - axis. Incorrect.

Answer:

\((1, 2)\), \((4, 2)\), and \((1, 4)\) (the third option in the given set of options)