Question

exercise #3: triangle efg is shown below with vertices at e(1,2), f(-3,8), and g(4,3).
(a) rotate \\( \triangle efg \\) by \\( 90^{\circ} \\) clockwise about the origin. label the image \\( \triangle efg \\). show where each vertex gets mapped below.
(b) based on (a), give an algebraic rule for a \\( 90^{\circ} \\) clockwise rotation about the origin.
(c) rotate \\( \triangle efg \\) by \\( 90^{\circ} \\) counterclockwise about the origin. label the image \\( \triangle efg \\). show where each vertex gets mapped below.
(d) based on (c), give an algebraic rule for a \\( 90^{\circ} \\) (e) use tracing paper to verify that \\( \triangle efg \\) is counterclockwise rotation about the origin. congruent to both \\( \triangle efg \\) and \\( \triangle efg \\).
exercise #4: indigo took a photo on her phone of the side of a house, but the image was incorrectly rotated. indigo placed her photo on a coordinate grid (shown on the left). point e is the top of the roof of the house.
(a) in what direction (c.w. or c.c.w.) should indigo rotate her image so that it is oriented with point e at the top?
(b) show where each of the five labeled points would get mapped using the appropriate rule from exercise #3.
\\( a(0,6) \
ightarrow \\)
\\( b(0,-6) \
ightarrow \\)
\\( c(3,7) \
ightarrow \\)
\\( d(3,-8) \
ightarrow \\)
\\( e(8,2) \
ightarrow \\)
(c) draw the rotated image using your results from (b) on the blank grid above.

Explanation:

Step1: Determine rotation direction

Since the image needs to be oriented with point \(E\) at the top, and considering the general orientation of coordinate - grid - based rotations, a \(90^{\circ}\) clockwise rotation (c.w.) will re - position the points such that \(E\) is at the top.

Step2: Apply \(90^{\circ}\) clockwise rotation rule

The rule for a \(90^{\circ}\) clockwise rotation about the origin is \((x,y)\to(y, - x)\).

• For \(A(0,6)\):

Substitute \(x = 0\) and \(y = 6\) into the rule \((x,y)\to(y, - x)\). We get \((6,0)\).

• For \(B(0, - 6)\):

Substitute \(x = 0\) and \(y=-6\) into the rule \((x,y)\to(y, - x)\). We get \((-6,0)\).

• For \(C(3,7)\):

Substitute \(x = 3\) and \(y = 7\) into the rule \((x,y)\to(y, - x)\). We get \((7,-3)\).

• For \(D(3, - 8)\):

Substitute \(x = 3\) and \(y=-8\) into the rule \((x,y)\to(y, - x)\). We get \((-8,-3)\).

• For \(E(8,2)\):

Substitute \(x = 8\) and \(y = 2\) into the rule \((x,y)\to(y, - x)\). We get \((2,-8)\).

Answer:

(a) c.w.
(b) \(A(0,6)\to(6,0)\), \(B(0, - 6)\to(-6,0)\), \(C(3,7)\to(7,-3)\), \(D(3, - 8)\to(-8,-3)\), \(E(8,2)\to(2,-8)\)