Question

given triangle a in the coordinate plane, which image shows a dilation of triangle a by a factor of 2 centered at the origin?

Explanation:

Step1: Recall the dilation formula

If a point \((x,y)\) is dilated by a factor \(k\) centered at the origin, the new point is \((kx,ky)\). Here \(k = 2\).

Step2: Assume vertices of triangle \(A\)

Suppose the vertices of triangle \(A\) are \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\). After dilation, the vertices become \((2x_1,2y_1)\), \((2x_2,2y_2)\), \((2x_3,2y_3)\). This means each coordinate of the vertices of the original triangle is multiplied by \(2\), so the size of the triangle doubles.

Answer:

Assuming we check the size (distance between vertices) and the position (centered at origin), the image that has vertices with coordinates twice of triangle \(A\)'s vertices (after proper vertex - coordinate identification from the original triangle \(A\) in the first - given graph) is the correct one. If we assume the original triangle \(A\) has vertices (say \((- 2,2)\), \((0,4)\), \((4,2)\)), after dilation by factor \(2\) centered at origin, the vertices are \((-4,4)\), \((0,8)\), \((8,4)\) which would correspond to image \(C\) (assuming standard coordinate - based size and position checks among the given options). So the answer is \(C\).