Question

triangle efg is drawn on the coordinate plane. the triangle is dilated by a scale factor of 1/3, with the center of dilation at the origin, to form triangle efg. then, triangle efg is translated 3 units to the right to form triangle efg. what is the coordinate of e?

Explanation:

Step1: Find the coordinate of \( E \)

Assume the coordinate of \( E \) is \( ( - 6,3) \) (from the coordinate - plane in the image).

Step2: Dilation operation

When a point \( (x,y) \) is dilated by a scale factor \( k=\frac{1}{3} \) with the center of dilation at the origin \((0,0)\), the new coordinate \((x',y')\) is given by the formula \( (x',y')=(kx,ky) \).
For point \( E(-6,3) \), after dilation: \( x'=\frac{1}{3}\times(-6)= - 2 \), \( y'=\frac{1}{3}\times3 = 1 \). So the coordinate of \( E'\) is \( (-2,1) \).

Step3: Translation operation

When a point \((x',y')\) is translated \( h = 3 \) units to the right (the translation rule for a horizontal translation to the right is \((x',y')\to(x'+h,y')\)).
For point \( E'(-2,1) \), after translation \( x''=-2 + 3=1 \), \( y'' = 1 \).

Answer:

\((1,1)\)