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 reflecti 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

• For \(AB\): Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), with \(A(-4,4)\) and \(B(-4,-2)\), we have \(AB=\sqrt{(-4+4)^2+(-2 - 4)^2}=\sqrt{0+(-6)^2}=6\)
• For \(BC\): Using \(B(-4,-2)\) and \(C(2,-2)\), \(BC=\sqrt{(2 + 4)^2+(-2+2)^2}=\sqrt{6^2+0}=6\)
• For \(AC\): Using \(A(-4,4)\) and \(C(2,-2)\), \(AC=\sqrt{(2 + 4)^2+(-2 - 4)^2}=\sqrt{36 + 36}=\sqrt{72}=6\sqrt{2}\)

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

• For \(A'B'\): Using \(A'(-2,2)\) and \(B'(-2,-1)\), \(A'B'=\sqrt{(-2+2)^2+(-1 - 2)^2}=\sqrt{0+(-3)^2}=3\)
• For \(B'C'\): Using \(B'(-2,-1)\) and \(C'(1,-1)\), \(B'C'=\sqrt{(1 + 2)^2+(-1+1)^2}=\sqrt{3^2+0}=3\)
• For \(A'C'\): Using \(A'(-2,2)\) and \(C'(1,-1)\), \(A'C'=\sqrt{(1 + 2)^2+(-1 - 2)^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}\)

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 \(2\), triangle \(ABC\) and \(A'B'C'\) are similar. And the transformation from \(ABC\) to \(A'B'C'\) is a dilation (a similarity transformation that changes the size of a figure).

Answer:

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