Question

original coordinates → final coordinates
d(2,3)→d(6,8)
e(3,−1)→e(9,−3)
f(−4,−2)→f(−12,−6)
(b) choose the general rule below that describes the dilation mapping △def to △def.
(x,y)→(1/3x,3y) (x,y)→(3x,1/3y)
(x,y)→(1/3y,1/3x) (x,y)→(3x,y)
(x,y)→(3x,3y) (x,y)→(3y,3x)
(x,y)→(1/3x,1/3y) (x,y)→(x,3y)

Explanation:

Step1: Analyze the x - coordinate of \(D\) and \(D'\)

Original \(x\) - coordinate of \(D\) is \(x = 2\), final \(x\) - coordinate of \(D'\) is \(x'=6\). Since \(6 = 3\times2\).

Step2: Analyze the y - coordinate of \(D\) and \(D'\)

Original \(y\) - coordinate of \(D\) is \(y = 3\), final \(y\) - coordinate of \(D'\) is \(y'=9\). Since \(9 = 3\times3\).

Step3: Analyze the x - coordinate of \(E\) and \(E'\)

Original \(x\) - coordinate of \(E\) is \(x = 3\), final \(x\) - coordinate of \(E'\) is \(x'=9\). Since \(9 = 3\times3\).

Step4: Analyze the y - coordinate of \(E\) and \(E'\)

Original \(y\) - coordinate of \(E\) is \(y=-1\), final \(y\) - coordinate of \(E'\) is \(y'=-3\). Since \(-3 = 3\times(-1)\).

Step5: Analyze the x - coordinate of \(F\) and \(F'\)

Original \(x\) - coordinate of \(F\) is \(x=-4\), final \(x\) - coordinate of \(F'\) is \(x'=-12\). Since \(-12 = 3\times(-4)\).

Step6: Analyze the y - coordinate of \(F\) and \(F'\)

Original \(y\) - coordinate of \(F\) is \(y = - 2\), final \(y\) - coordinate of \(F'\) is \(y'=-6\). Since \(-6 = 3\times(-2)\).

For a general point \((x,y)\) in \(\triangle DEF\), after dilation, the point \((x',y')\) in \(\triangle D'E'F'\) satisfies \(x'=3x\) and \(y' = 3y\).

Answer:

\((x,y)\to(3x,3y)\)