Question

rotations
triangle hop has coordinates h(2, 1), o(-3, 4), and p(5, 7). determine the coordinates of the image of \\(\triangle hop\\) after each rotation.

1. a rotation 90° clockwise about the origin
2. a rotation 90° counter - clockwise about the origin
3. a rotation 180° about the origin
4. fill in the blank 2 points

h(
1
2
)
p(
-5
-7
)
o(
3
4
)
h(
-2
-1
)

Explanation:

Step1: Recall 90 - degree clockwise rotation rule

For a point \((x,y)\) rotated 90 - degree clockwise about the origin, the new coordinates \((x',y')\) are given by \((y,-x)\).
For \(H(2,1)\), \(x = 2\) and \(y = 1\), so \(H'=(1,-2)\).
For \(O(-3,4)\), \(x=-3\) and \(y = 4\), so \(O'=(4,3)\).
For \(P(5,7)\), \(x = 5\) and \(y = 7\), so \(P'=(7,-5)\).

Step2: Recall 90 - degree counter - clockwise rotation rule

For a point \((x,y)\) rotated 90 - degree counter - clockwise about the origin, the new coordinates \((x',y')\) are given by \((-y,x)\).
For \(H(2,1)\), \(x = 2\) and \(y = 1\), so \(H'=(-1,2)\).
For \(O(-3,4)\), \(x=-3\) and \(y = 4\), so \(O'=(-4,-3)\).
For \(P(5,7)\), \(x = 5\) and \(y = 7\), so \(P'=(-7,5)\).

Step3: Recall 180 - degree rotation rule

For a point \((x,y)\) rotated 180 - degree about the origin, the new coordinates \((x',y')\) are given by \((-x,-y)\).
For \(H(2,1)\), \(x = 2\) and \(y = 1\), so \(H'=(-2,-1)\).
For \(O(-3,4)\), \(x=-3\) and \(y = 4\), so \(O'=(3,-4)\).
For \(P(5,7)\), \(x = 5\) and \(y = 7\), so \(P'=(-5,-7)\).

Answer:

For a 90 - degree clockwise rotation: \(H'(1,-2)\), \(O'(4,3)\), \(P'(7,-5)\)
For a 90 - degree counter - clockwise rotation: \(H'(-1,2)\), \(O'(-4,-3)\), \(P'(-7,5)\)
For a 180 - degree rotation: \(H'(-2,-1)\), \(O'(3,-4)\), \(P'(-5,-7)\)