Question

the vertices of $\triangle abc$ are $a(-5,5)$, $b(-2,5)$, and $c(-3,3)$. if $\triangle abc$ is reflected across the line $y = -2$ to produce the image $\triangle abc$, find the coordinates of the vertex $c$.

the coordinates of $c$ after a reflection across the line $y = -2$ are $\square$.
(type an ordered pair)

Explanation:

Step1: Recall reflection over horizontal line

When reflecting a point \((x,y)\) over the horizontal line \(y = k\), the \(x\)-coordinate remains the same, and the \(y\)-coordinate changes. The formula for the reflection of a point \((x,y)\) over the line \(y = k\) is \((x, 2k - y)\). Here, \(k=-2\) and the point \(C\) is \((-3,3)\).

Step2: Apply the reflection formula

For point \(C(-3,3)\) and \(k = - 2\), substitute into the formula. The \(x\)-coordinate remains \(-3\). For the \(y\)-coordinate, calculate \(2k - y\). Substitute \(k=-2\) and \(y = 3\): \(2\times(-2)-3=-4 - 3=-7\).

Answer:

\((-3, - 7)\)