Question

triangle abc has been reflected over the x - axis to create triangle abc. which of the following statements is true?
bc = 2
ab = 2
bc = 2
ac = 2

Explanation:

Step1: Recall reflection property

A reflection over the x - axis is a rigid transformation. Rigid transformations preserve distance. That is, the distance between any two points in the pre - image is equal to the distance between the corresponding two points in the image. For a point \(P(x,y)\) reflected over the x - axis to \(P'(x, - y)\), if we consider two points \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the pre - image and their corresponding points \(A'(x_1,-y_1)\) and \(B'(x_2,-y_2)\) in the image, the distance \(d(A,B)=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) and \(d(A',B')=\sqrt{(x_2 - x_1)^2+(-y_2 + y_1)^2}=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

Step2: Analyze the segments

The segment \(AA'\) is perpendicular to the x - axis. If we assume the coordinates of \(A=(x,y)\) and \(A'=(x, - y)\), the length of \(AA'\) is given by \(|y-(-y)|=|2y|\). From the graph, we can see that the distance between a point and its reflection over the x - axis (for example, between \(A\) and \(A'\)) is the vertical distance. If we assume the y - coordinate of \(A\) is 1 and of \(A'\) is - 1, then the length of \(AA'=|1-(-1)| = 2\).

Answer:

\(AA' = 2\)