Question

select the correct answer. how many triangles in the diagram can be mapped to one another by similarity transformations? a. 2 b. 4 c. 0 d. 3

Explanation:

Step1: Analyze triangle \(ABC\)

Assume \(A=(2,2)\), \(B=(8,8)\), \(C=(14,2)\).
Base \(AC = 14 - 2=12\), height (vertical distance from \(B\) to \(AC\)) \(=8 - 2 = 6\).

Step2: Analyze triangle \(DEF\)

Assume \(D=(16,5)\), \(E=(22,9)\), \(F=(28,5)\).
Base \(DF=28 - 16 = 12\), height (vertical distance from \(E\) to \(DF\)) \(=9 - 5=4\).

Step3: Analyze triangle \(PQR\)

Assume \(P=(12,11)\), \(Q=(22,17)\), \(R=(28,11)\).
Base \(PR = 28 - 12=16\), height (vertical distance from \(Q\) to \(PR\)) \(=17 - 11 = 6\).

Step4: Analyze triangle \(GHI\)

Assume \(G=(0,20)\), \(H=(16,19)\), \(I=(8,12)\).
Base \(GH\) (using distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), \(x_1 = 0,y_1 = 20,x_2=16,y_2 = 19\), \(d_{GH}=\sqrt{(16 - 0)^2+(19 - 20)^2}=\sqrt{256 + 1}=\sqrt{257}\), base \(GI\) (\(x_1 = 0,y_1 = 20,x_2 = 8,y_2 = 12\), \(d_{GI}=\sqrt{(8 - 0)^2+(12 - 20)^2}=\sqrt{64 + 64}=\sqrt{128}\), base \(HI\) (\(x_1 = 16,y_1 = 19,x_2 = 8,y_2 = 12\), \(d_{HI}=\sqrt{(8 - 16)^2+(12 - 19)^2}=\sqrt{64+49}=\sqrt{113}\))

Similarity transformations (dilations, rotations, reflections, translations) require that the ratios of corresponding side - lengths are equal.
For two triangles \(\triangle_1\) and \(\triangle_2\) with side - lengths \(a_1,b_1,c_1\) and \(a_2,b_2,c_2\) respectively, \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\) (for non - degenerate triangles).
By checking the ratios of side - lengths (using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) for all pairs of triangles:

• For \(\triangle ABC\) and \(\triangle DEF\): Ratios of base and height \(\frac{12}{12}

eq\frac{6}{4}\)

• For \(\triangle ABC\) and \(\triangle PQR\): \(\frac{12}{16}

eq\frac{6}{6}\)

• For \(\triangle ABC\) and \(\triangle GHI\): Using side - length ratios (calculated via distance formula) not equal.
• For \(\triangle DEF\) and \(\triangle PQR\): \(\frac{12}{16}

eq\frac{4}{6}\)

• For \(\triangle DEF\) and \(\triangle GHI\): Side - length ratios (via distance formula) not equal.
• For \(\triangle PQR\) and \(\triangle GHI\): Side - length ratios (via distance formula) not equal.

Answer:

C. \(0\)