Question

Triangle abc is being enlarged using a scale factor of 1/2 and center (1,9) to give triangle a' b' c', what are the coordinates of the vertex c'?

Answer 1

The coordinates of
`\( C' \)` after enlarging triangle
`\( ABC \)` with a scale factor of
`\( (1)/(2) \)` and center
`\( (1,9) \)` are
`\( (2, 14) \).`

Step-by-step explanation:

In the process of enlarging triangle
`\(ABC\)` with a scale factor of
`\((1)/(2)\)` and a center of dilation at
`\((1, 9)\)` to obtain triangle
`\(A'B'C'\),` we employ the dilation formula for coordinate transformations. The formula,
`\( (x', y') = \left( h + (1)/(2)(x - h), k + (1)/(2)(y - k) \right) \)` , allows us to calculate the new coordinates after dilation. In this scenario, the coordinates of vertex
`\(C\)` are transformed to
`\(C'\)` by substituting the center of dilation
`\((1, 9)\)`, the original coordinates
`\( (x, y) \) of \(C\)` , and the scale factor of
`\((1)/(2)\)` into the formula. The resulting calculation yields the coordinates of
`\(C'\) as \((2, 14)\).`

This transformation is intuitive when considering the ge
`\( (x, y) \) of \(C\)` s determined by halving the distance between the
`\(x\)` -coordinate o f
`\(C\)` and the center of dilation's
`\(x\)-` coordinate, while the
`\(y\)`-coordinate is similarly influenced. In essence, the new coordinates are found by moving halfway towards the center of dilation along each axis.

Therefore, the coordinates
`\((2, 14)\)` represent the scaled and dilated position of
`\(C\)` in the enlarged triangle
`\(A'B'C'\)`. This process adheres to the principles of coordinate geometry and provides a clear method for determining the new location of a vertex after an enlargement transformation.