Question

which sequence of transformations shows that figure p is congruent to figure q?
reflecting figure p over the y - axis and translating the resulting figure down 3 units
reflecting figure p over the x - axis and translating the resulting figure 7 units to the right
reflecting figure p over the x - axis and rotating the resulting figure 180° about the origin
translating figure p 1 unit to the right and rotating the resulting figure 180° about the origin
translating figure p 4 units to the right and rotating the resulting figure 180° about the origin

Explanation:

Step1: Analyze reflection over \(x\) - axis

When a figure is reflected over the \(x\) - axis, the \(x\) - coordinates of its points remain the same, and the \(y\) - coordinates are multiplied by \(- 1\). If we reflect figure \(P\) over the \(x\) - axis, we get a figure that is upside - down compared to \(P\).

Step2: Analyze translation

After reflecting figure \(P\) over the \(x\) - axis, we need to move it horizontally. The \(x\) - coordinate of the vertex of figure \(P\) (before reflection) is \(x=-3\), and the \(x\) - coordinate of the vertex of figure \(Q\) is \(x = 4\). The distance between \(-3\) and \(4\) is \(|4-(-3)|=7\). So, after reflecting figure \(P\) over the \(x\) - axis, translating the resulting figure \(7\) units to the right will map the reflected figure to figure \(Q\).

Let's check other options:

• For reflection over \(y\) - axis: Reflecting over \(y\) - axis changes \(x\) to \(-x\). Then translating down \(3\) units won't map \(P\) to \(Q\).
• For reflection over \(x\) - axis and \(180^{\circ}\) rotation: A \(180^{\circ}\) rotation about the origin \((x,y)\to(-x,-y)\) after reflection over \(x\) - axis \((x,y)\to(x, - y)\) gives \((-x,y)\) which is not the transformation needed.
• For translating \(1\) unit right and \(180^{\circ}\) rotation: \(180^{\circ}\) rotation about the origin \((x,y)\to(-x,-y)\) after translation \((x,y)\to(x + 1,y)\) gives \((-(x + 1),-y)\) which is not correct.
• For translating \(4\) units right and \(180^{\circ}\) rotation: \(180^{\circ}\) rotation about the origin \((x,y)\to(-x,-y)\) after translation \((x,y)\to(x + 4,y)\) gives \((-(x + 4),-y)\) which is not correct.

Answer:

Reflecting figure \(P\) over the \(x\) - axis and translating the resulting figure \(7\) units to the right.