Question

write the ordered pair location of the new point after the given transformations.
translate 4 units down, rotate 180° counterclockwise about the origin, then translate 5 units right
point c(-4, -5)
point c: ( , )

Explanation:

Step1: Translate 4 units down

To translate a point \((x, y)\) \(k\) units down, we subtract \(k\) from the \(y\)-coordinate. For point \(C(-4, -5)\), translating 4 units down gives \((-4, -5 - 4) = (-4, -9)\).

Step2: Rotate \(180^\circ\) counterclockwise about the origin

The rule for a \(180^\circ\) rotation about the origin is \((x, y) \to (-x, -y)\). Applying this to \((-4, -9)\), we get \((-(-4), -(-9)) = (4, 9)\).

Step3: Translate 5 units right

To translate a point \((x, y)\) \(k\) units right, we add \(k\) to the \(x\)-coordinate. Applying this to \((4, 9)\), we get \((4 + 5, 9) = (9, 9)\)? Wait, no, wait. Wait, let's re - check step 2. Wait, the point after step 1 is \((-4, -9)\). Rotating \(180^\circ\) counterclockwise about the origin: the formula is \((x,y)\to(-x,-y)\). So \(x=-4\), so \(-x = 4\); \(y = -9\), so \(-y=9\). Then step 3: translate 5 units right. So \(x\)-coordinate: \(4 + 5=9\), \(y\)-coordinate remains 9? But wait, maybe I made a mistake. Wait, let's start over.

Wait, original point \(C(-4,-5)\)

Step 1: Translate 4 units down. The translation rule for down is \((x,y)\to(x,y - 4)\). So \((-4,-5-4)=(-4,-9)\)

Step 2: Rotate \(180^\circ\) counterclockwise about the origin. The rotation rule for \(180^\circ\) is \((x,y)\to(-x,-y)\). So for \((-4,-9)\), we have \((-(-4),-(-9))=(4,9)\)

Step 3: Translate 5 units right. The translation rule for right is \((x,y)\to(x + 5,y)\). So \((4+5,9)=(9,9)\). But the options have 9 and - 9. Wait, maybe I messed up the rotation direction? Wait, \(180^\circ\) counterclockwise and \(180^\circ\) clockwise are the same. Wait, maybe the problem is in the first step. Wait, let's check the problem again.

Wait, the problem says: Translate 4 units down, rotate \(180^\circ\) counterclockwise about the origin, then translate 5 units right.

Wait, maybe I made a mistake in the rotation. Wait, the point after translation is \((-4,-9)\). Rotating \(180^\circ\) about the origin: the formula is \((x,y)\to(-x,-y)\). So \((-4,-9)\) becomes \((4,9)\). Then translating 5 units right: \(4 + 5=9\), \(y = 9\). But the options have 9 and - 9. Wait, maybe the original point was \((-4,-5)\), translate 4 units down: \((-4,-5 - 4)=(-4,-9)\). Rotate \(180^\circ\): \((4,9)\). Translate 5 units right: \((9,9)\). But the options have 9 and - 9. Wait, maybe I messed up the rotation. Wait, no, \(180^\circ\) rotation: \((x,y)\to(-x,-y)\). So if the point is \((x,y)\), after \(180^\circ\) rotation, it's \((-x,-y)\). So \((-4,-9)\) becomes \((4,9)\). Then translate 5 units right: \((9,9)\). But the options include 9 and - 9. Wait, maybe there is a miscalculation. Wait, let's check the steps again.

Wait, maybe the user made a typo, or maybe I misread the original point. The original point is \(C(-4,-5)\).

Step 1: Translate 4 units down: \((-4,-5-4)=(-4,-9)\)

Step 2: Rotate \(180^\circ\) counterclockwise about the origin: \((-(-4),-(-9))=(4,9)\)

Step 3: Translate 5 units right: \((4 + 5,9)=(9,9)\)

But the options have 9 and - 9. Wait, maybe the rotation is \(180^\circ\) clockwise? But \(180^\circ\) clockwise and counterclockwise are the same. Wait, maybe the problem is in the translation direction. Wait, "translate 4 units down" is correct. "Translate 5 units right" is correct.

Wait, maybe I made a mistake in the rotation. Wait, the formula for \(180^\circ\) rotation is \((x,y)\to(-x,-y)\). So if the point is \((x,y)\), after rotation, it's \((-x,-y)\). So for \((-4,-9)\), \(-x = 4\), \(-y = 9\). So that's correct. Then translating 5 units right: \(4+5 = 9\), \(y = 9\). So the ordered pair is \((9,9)\). But the options have 9 and - 9. Wait, maybe the original point was \((-4,5)\)? No, the original point is \((-4,-5)\).

Wait, maybe the rotation is \(180^\circ\) about the origin, but the rule is \((x,y)\to(-x,-y)\). So if the point is \((-4,-9)\), then \(-x = 4\), \(-y = 9\). Then translate 5 units right: \(4 + 5=9\), \(y = 9\). So the answer should be \((9,9)\). But let's check the options. The options are 1, - 1, 5, - 5, 4, - 4, 9, - 9. So 9 is there. So the ordered pair is \((9,9)\)? Wait, but maybe I made a mistake in step 1. Wait, translate 4 units down: \(y\)-coordinate: \(-5-4=-9\), correct. Step 2: rotate \(180^\circ\): \((-4,-9)\to(4,9)\), correct. Step 3: translate 5 units right: \(4 + 5 = 9\), \(y = 9\). So the answer is \((9,9)\).

Answer:

\((9,9)\)