Question

use the information given about triangle abc. the coordinates of triangle abc are a (-4, 4), b (-4, -2), and c (2, -2). the coordinates of triangle abc are a(-2, 2), b(-2, -1), and c(1, -1). which of the following statements is true? a triangle abc is similar to triangle abc because a dilation was performed on triangle abc. b the two triangles are congruent because a reflection was performed on triangle abc. c triangle abc is similar because a rotation was performed on triangle abc. d the two triangles are congruent because a translation was performed on triangle abc.

Explanation:

Step1: Calculate the side lengths of triangle \(ABC\)

Using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)
For \(AB\): \(x_1=-4,y_1 = 4,x_2=-4,y_2=-2\), \(AB=\sqrt{(-4+4)^2+(-2 - 4)^2}=\sqrt{0 + 36}=6\)
For \(BC\): \(x_1=-4,y_1=-2,x_2=2,y_2=-2\), \(BC=\sqrt{(2 + 4)^2+(-2+2)^2}=\sqrt{36+0}=6\)
For \(AC\): \(x_1=-4,y_1 = 4,x_2=2,y_2=-2\), \(AC=\sqrt{(2 + 4)^2+(-2 - 4)^2}=\sqrt{36 + 36}=\sqrt{72}=6\sqrt{2}\)

Step2: Calculate the side lengths of triangle \(A'B'C'\)

For \(A'B'\): \(x_1=-2,y_1 = 2,x_2=-2,y_2=-1\), \(A'B'=\sqrt{(-2+2)^2+(-1 - 2)^2}=\sqrt{0+9}=3\)
For \(B'C'\): \(x_1=-2,y_1=-1,x_2=1,y_2=-1\), \(B'C'=\sqrt{(1 + 2)^2+(-1+1)^2}=\sqrt{9+0}=3\)
For \(A'C'\): \(x_1=-2,y_1 = 2,x_2=1,y_2=-1\), \(A'C'=\sqrt{(1 + 2)^2+(-1 - 2)^2}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}\)

Step3: Check the ratio of side lengths

\(\frac{AB}{A'B'}=\frac{6}{3}=2\), \(\frac{BC}{B'C'}=\frac{6}{3}=2\), \(\frac{AC}{A'C'}=\frac{6\sqrt{2}}{3\sqrt{2}}=2\)

Since the ratio of corresponding side lengths is constant (\(k = 2\)), and dilation changes the size of a figure by a scale factor while keeping the shape (corresponding angles equal). Reflection, rotation and translation are rigid - motion transformations (they preserve the size and shape of the figure, i.e., congruence). Here the size of the triangle has changed (side lengths of \(A'B'C'\) are half of \(ABC\)’s side lengths), so it is a dilation.

Answer:

A. Triangle \(ABC\) is similar to triangle \(A'B'C'\) because a dilation was performed on triangle \(ABC\)