Question

4.1 reflections · spiral review
graph the image of each figure under the given reflection. determine the coordinates of the image.

1. △abc in the line ( y = x )
2. trapezoid defg in the line ( x = - 1 )
3. parallelogram rstu in the line ( y = x )
4. square klmn in the line ( y = - 2 )
5. determine the coordinates of ( s ( - 7,1 ) ) after a reflection in the line ( y = 3 ).
6. determine the coordinates of ( q ( 6, - 4 ) ) after a reflection in the line ( x = 2 ).

Explanation:

Step1: Reflection rule for \(y = x\)

The rule for reflecting a point \((a,b)\) over the line \(y=x\) is \((a,b)\to(b,a)\).
For \(\triangle ABC\), assume \(A(-4,2)\), \(B(-1,1)\), \(C(-2,-3)\)

• For point \(A(-4,2)\):

\((-4,2)\to(2,-4)\)

• For point \(B(-1,1)\):

\((-1,1)\to(1,-1)\)

• For point \(C(-2,-3)\):

\((-2,-3)\to(-3,-2)\)

Step2: Reflection rule for \(x=-1\)

The rule for reflecting a point \((a,b)\) over the line \(x = c\) is \((a,b)\to(2c - a,b)\). Here \(c=-1\)
For trapezoid \(DEFG\), assume \(D(-1,-3)\), \(E(-1,2)\), \(F(2,2)\), \(G(4,-3)\)

• For point \(D(-1,-3)\):

\((-1,-3)\to(-1,-3)\) (since it lies on \(x = - 1\))

• For point \(E(-1,2)\):

\((-1,2)\to(-1,2)\) (since it lies on \(x=-1\))

• For point \(F(2,2)\):

\(2c-a=2\times(-1)-2=-4\), so \((2,2)\to(-4,2)\)

• For point \(G(4,-3)\):

\(2c - a=2\times(-1)-4=-6\), so \((4,-3)\to(-6,-3)\)

Step3: Reflection rule for \(y = 3\) (for problem 5)

The rule for reflecting a point \((a,b)\) over the line \(y = k\) is \((a,b)\to(a,2k - b)\). Here \(k = 3\)
For \(S(-7,1)\):
\(2k - b=2\times3-1 = 5\), so \((-7,1)\to(-7,5)\)

Step4: Reflection rule for \(x = 2\) (for problem 6)

The rule for reflecting a point \((a,b)\) over the line \(x=c\) is \((a,b)\to(2c - a,b)\). Here \(c = 2\)
For \(Q(6,-4)\):
\(2c - a=2\times2-6=-2\), so \((6,-4)\to(-2,-4)\)

Answer:

1. \(A'(2,-4)\), \(B'(1,-1)\), \(C'(-3,-2)\)
2. \(D'(-1,-3)\), \(E'(-1,2)\), \(F'(-4,2)\), \(G'(-6,-3)\)
3. (Using \(R(-3,1)\), \(S(2,3)\), \(T(3,-2)\), \(U(-2,-4)\) assume coordinates from graph) \(R'(1,-3)\), \(S'(3,2)\), \(T'(-2,3)\), \(U'(-4,-2)\)
4. (Using \(K(-1,-1)\), \(L(-3,1)\), \(M(-1,3)\), \(N(1,1)\) assume coordinates from graph) \(K'(-1,-3)\), \(L'(-3,-5)\), \(M'(-1,-7)\), \(N'(1,-5)\)
5. \((-7,5)\)
6. \((-2,-4)\)