Question

find the dilation of the quadrilateral with the center at the origin and scale factor of 2.

Explanation:

Step1: Recall dilation rule

For a dilation with center at the origin \((0,0)\) and scale - factor \(k\), if a point \((x,y)\) is dilated, the new point \((x',y')\) is given by \((x',y')=(k\cdot x,k\cdot y)\).

Step2: Identify vertices of quadrilateral

Assume the vertices of quadrilateral \(RSTU\) are \(R(x_1,y_1)\), \(S(x_2,y_2)\), \(T(x_3,y_3)\), \(U(x_4,y_4)\). Let's say \(R(- 2,-2)\), \(S(-1,1)\), \(T(0,1)\), \(U(2,-1)\) (by observing the grid).

Step3: Apply dilation formula

For point \(R(-2,-2)\), with \(k = 2\), the new point \(R'=(2\times(-2),2\times(-2))=(-4,-4)\).
For point \(S(-1,1)\), the new point \(S'=(2\times(-1),2\times1)=(-2,2)\).
For point \(T(0,1)\), the new point \(T'=(2\times0,2\times1)=(0,2)\).
For point \(U(2,-1)\), the new point \(U'=(2\times2,2\times(-1))=(4,-2)\).

Answer:

The vertices of the dilated quadrilateral are \(R'(-4,-4)\), \(S'(-2,2)\), \(T'(0,2)\), \(U'(4,-2)\)