Question

triangle pqr is reflected across the line y = -4. the coordinates of the vertices of the image of △pqr are p(-8, -9), q(-3, -7), and r(1, -10). what are the coordinates of r? a (1, -10) b (1, 2) c (-13, 2) d (-9, -10)

Explanation:

Step1: Recall Reflection Over Horizontal Line

When a point \((x,y)\) is reflected over the line \(y = k\), the \(x\)-coordinate remains the same, and the \(y\)-coordinate satisfies the midpoint formula: the midpoint between \(y\) (original) and \(y'\) (image) is \(k\). So, \(\frac{y + y'}{2}=k\), which rearranges to \(y = 2k - y'\). Here, \(k=-4\) and we know \(R'=(1, - 10)\), so we need to find the original \(y\)-coordinate of \(R\) using \(y = 2(-4)-(-10)\).

Step2: Calculate Original \(y\)-Coordinate

First, compute \(2(-4)-(-10)\). \(2(-4)=-8\), and \(-(-10) = 10\). Then, \(-8 + 10=2\). The \(x\)-coordinate of \(R\) is the same as the \(x\)-coordinate of \(R'\) because reflection over a horizontal line doesn't change the \(x\)-coordinate. So the \(x\)-coordinate is \(1\). Thus, the coordinates of \(R\) are \((1,2)\).

Answer:

B. \((1, 2)\)