Question

1. cornell is designing a character for a computer game. he decides to lay out the design on a coordinate grid as shown below. cornell would like to add one additional shape to his character by rotating kite ( a b c d ) by ( 90^{circ} ) clockwise about the origin. (a) why would this rotation leave point ( a ) where it is? (b) give the image points for where each of the following is mapped under this rotation: ( b(-7,1)

ightarrow ) ( c(-10,5)
ightarrow ) ( d(-5,5)
ightarrow ) (c) draw in and shade kite ( a^{prime} b^{prime} c^{prime} d^{prime} ). reasoning 8. are kites ( a b c d ) and ( a^{prime} b^{prime} c^{prime} d^{prime} ) symmetric across the ( y )-axis? what can you recall about line symmetry?

Explanation:

Step1: Analyze why point A remains

Point A is at the origin \((0,0)\). When rotating a point \((x,y)\) about the origin by \(90^{\circ}\) clockwise, the formula is \((x,y)\to(y, -x)\). For \(A(0,0)\), substituting into the formula: \(x = 0,y = 0\), so \((0,0)\to(0,0)\).

Step2: Find the image of point B

For \(B(-7,1)\), using the \(90^{\circ}\) clockwise rotation formula \((x,y)\to(y, -x)\). Here \(x=-7,y = 1\), then \(B(-7,1)\to(1,7)\).

Step3: Find the image of point C

For \(C(-10,5)\), using the formula \((x,y)\to(y, -x)\). Here \(x=-10,y = 5\), then \(C(-10,5)\to(5,10)\).

Step4: Find the image of point D

For \(D(-5,5)\), using the formula \((x,y)\to(y, -x)\). Here \(x=-5,y = 5\), then \(D(-5,5)\to(5,5)\).

Two figures are symmetric across the \(y\) - axis if for every point \((x,y)\) in one figure, the point \((-x,y)\) is in the other figure.
For kite \(ABCD\) with points \(B(-7,1)\), \(C(-10,5)\), \(D(-5,5)\) and kite \(A'B'C'D'\) with \(B'(1,7)\), \(C'(5,10)\), \(D'(5,5)\).
Take \(B(-7,1)\) and \(B'(1,7)\): \(-x=-(-7) = 7
eq1\) (for \(y = 1\) in \(B\) and \(y = 7\) in \(B'\)).
Take \(C(-10,5)\) and \(C'(5,10)\): \(-x=-(-10)=10
eq5\) (for \(y = 5\) in \(C\) and \(y = 10\) in \(C'\)).
Take \(D(-5,5)\) and \(D'(5,5)\): \(-x=-(-5) = 5\) (but \(y\) - values are the same, but other points do not follow the \(y\) - axis symmetry rule \((x,y)\to(-x,y)\)).

Answer:

(a) Because point \(A\) is at the origin \((0,0)\), and rotating \((0,0)\) \(90^{\circ}\) clockwise about the origin using the formula \((x,y)\to(y,-x)\) gives \((0,0)\) (since \(x = 0,y = 0\), \(y=0,-x = 0\)).

(b) \(B(-7,1)\to(1,7)\), \(C(-10,5)\to(5,10)\), \(D(-5,5)\to(5,5)\)

(c) To draw \(A'B'C'D'\):

• Plot \(A(0,0)\) (remains the same).
• Plot \(B'(1,7)\), \(C'(5,10)\), \(D'(5,5)\)
• Connect the points \(A'\) to \(B'\) to \(C'\) to \(D'\) to \(A'\) and shade the region.

For question 8: