Question

triangle def is translated 6 units to the left and 8 units down. the result is △def, 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 translation. original coordinates → final coordinates d(2,5)→d□,□ e(1,8)→e□,□ f(5,4)→f□,□ (b) choose the general rule below that describes the translation mapping △def to △def. (x,y)→(-6x,-8y) (x,y)→(x - 8,y - 6) (x,y)→(8x,6y) (x,y)→(x - 6,y - 8) (x,y)→(x + 6,y + 8) (x,y)→(x + 8,y + 6) (x,y)→(-8x,-6y) (x,y)→(6x,8y)

Explanation:

Step1: Determine the translation rule for coordinates

When a point \((x,y)\) is translated \(a\) units to the left and \(b\) units down, the new coordinates \((x',y')\) are given by the rule \((x',y')=(x - a,y - b)\). Here \(a = 6\) and \(b=8\).

Step2: Calculate \(D'\)

For \(D(2,5)\), using the rule \((x - 6,y - 8)\):
\(x'=2-6=-4\), \(y'=5 - 8=-3\). So \(D(2,5)\to D'(-4,-3)\)

Step3: Calculate \(E'\)

For \(E(1,8)\), using the rule \((x - 6,y - 8)\):
\(x'=1-6=-5\), \(y'=8 - 8=0\). So \(E(1,8)\to E'(-5,0)\)

Step4: Calculate \(F'\)

For \(F(5,4)\), using the rule \((x - 6,y - 8)\):
\(x'=5-6=-1\), \(y'=4 - 8=-4\). So \(F(5,4)\to F'(-1,-4)\)

Step5: Verify the general rule

The general rule for a translation of \(6\) units to the left (\(x\) - coordinate: \(x-6\)) and \(8\) units down (\(y\) - coordinate: \(y - 8\)) is \((x,y)\to(x - 6,y - 8)\)

Answer:

(a)
\(D(2,5)\to D'(-4,-3)\)
\(E(1,8)\to E'(-5,0)\)
\(F(5,4)\to F'(-1,-4)\)

(b) \((x,y)\to(x - 6,y - 8)\)