Question

a dilation centered at the origin with a scale factor of 3 is applied to \\( \triangle x y z \\).
the result is \\( \triangle x ^ { \prime } y ^ { \prime } z ^ { \prime } \\), as shown below.
(a) the arrows below show that the coordinates on the left are
mapped to the coordinates on the right. fill in the blanks to
give the coordinates after the dilation.
original coordinates \\( \
ightarrow \\) final coordinates
\\( x ( - 1,2 ) \
ightarrow x ^ { \prime } ( \square, \square ) \\)
\\( y ( 4,1 ) \
ightarrow y ^ { \prime } ( \square, \square ) \\)
\\( z ( - 2, - 3 ) \
ightarrow z ^ { \prime } ( \square, \square ) \\)
(b) choose the general rule below that describes the dilation
mapping \\( \triangle x y z \\) to \\( \triangle x ^ { \prime } y ^ { \prime } z ^ { \prime } \\).
\\( ( x, y ) \
ightarrow ( 3 x, 3 y ) \\)
\\( ( x, y ) \
ightarrow \left( \frac { 1 } { 3 } x, \frac { 1 } { 3 } y \
ight) \\)
\\( ( x, y ) \
ightarrow \left( 3 x, \frac { 1 } { 3 } y \
ight) \\)
\\( ( x, y ) \
ightarrow \left( \frac { 1 } { 3 } x, 3 y \
ight) \\)
\\( ( x, y ) \
ightarrow ( 3 x, y ) \\)
\\( ( x, y ) \
ightarrow ( 3 y, 3 x ) \\)
\\( ( x, y ) \
ightarrow \left( \frac { 1 } { 3 } y, \frac { 1 } { 3 } x \
ight) \\)
\\( ( x, y ) \
ightarrow ( x, 3 y ) \\)

Explanation:

Step1: Apply dilation formula

For a dilation centered at the origin with scale factor \(k\), the formula is \((x,y)\to(kx,ky)\). Here \(k = 3\).
For point \(X(-1,2)\):
\(x=-1,y = 2\), after dilation \(x'=3\times(-1)=-3,y'=3\times2 = 6\). So \(X(-1,2)\to X'(-3,6)\)

Step2: Apply dilation formula for \(Y\)

For point \(Y(4,1)\):
\(x = 4,y=1\), after dilation \(x'=3\times4=12,y'=3\times1=3\). So \(Y(4,1)\to Y'(12,3)\)

Step3: Apply dilation formula for \(Z\)

For point \(Z(-2,-3)\):
\(x=-2,y=-3\), after dilation \(x'=3\times(-2)=-6,y'=3\times(-3)=-9\). So \(Z(-2,-3)\to Z'(-6,-9)\)

Step4: Determine the general rule

Since the formula \((x,y)\to(kx,ky)\) with \(k = 3\), the general rule is \((x,y)\to(3x,3y)\)

Answer:

(a) \(X(-1,2)\to X'(-3,6)\), \(Y(4,1)\to Y'(12,3)\), \(Z(-2,-3)\to Z'(-6,-9)\)
(b) \((x,y)\to(3x,3y)\)