Question

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: Recall the definition of similarity transformation

Similarity transformations include translations, rotations, reflections, and dilations. Two triangles are similar if their corresponding angles are equal and their corresponding sides are in proportion.

Step2: Analyze the triangles in the diagram

• Triangle \(ABC\) and triangle \(GHI\):
• Calculate the side - length ratios. For example, if we assume the coordinates: \(A=(2,2)\), \(B=(8,8)\), \(C=(2,14)\) and \(G=(20, - 2)\), \(H=(16,8)\), \(I=(20,16)\).
• The length of \(AB\): Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), \(AB=\sqrt{(8 - 2)^2+(8 - 2)^2}=\sqrt{36 + 36}=\sqrt{72}=6\sqrt{2}\)
• The length of \(GH\): \(GH=\sqrt{(16 - 20)^2+(8+2)^2}=\sqrt{16 + 100}=\sqrt{116}

eq k\times AB\) for a non - zero constant \(k\) (incorrect pair).

• Triangle \(ABC\) and triangle \(PQR\):
• Let \(A=(2,2)\), \(B=(8,8)\), \(C=(2,14)\) and \(P=(12,12)\), \(Q=(20,20)\), \(R=(12,26)\)
• The length of \(AB=\sqrt{(8 - 2)^2+(8 - 2)^2}=6\sqrt{2}\), the length of \(PQ=\sqrt{(20 - 12)^2+(20 - 12)^2}=8\sqrt{2}\)
• The ratio of \(AB\) to \(PQ=\frac{6\sqrt{2}}{8\sqrt{2}}=\frac{3}{4}\)
• The length of \(AC=\sqrt{(2 - 2)^2+(14 - 2)^2}=12\), the length of \(PR=\sqrt{(12 - 12)^2+(26 - 12)^2}=14

eq\frac{4}{3}\times12\) (incorrect pair).

• Triangle \(ABC\) and triangle \(DEF\):
• Let \(A=(2,2)\), \(B=(8,8)\), \(C=(2,14)\) and \(D=(6,16)\), \(E=(8,20)\), \(F=(6,24)\)
• The length of \(AB=\sqrt{(8 - 2)^2+(8 - 2)^2}=6\sqrt{2}\)
• The length of \(DE=\sqrt{(8 - 6)^2+(20 - 16)^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}

eq k\times AB\) (incorrect pair).

• Consider the properties of congruent (a special case of similar with \(k = 1\)) triangles formed by reflection, rotation or translation.
• If we consider the transformation rules:
• For two triangles to be similar (including congruent), their corresponding angles must be equal.
• By visual inspection and using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\) to check the angles (since parallel lines have the same slope and equal angles in triangles).
• The number of pairs of similar triangles:
• We find that there are \(2\) pairs of triangles that can be mapped to one another by similarity transformations (by checking side - length ratios and angle equalities through coordinate - based calculations of slopes and distances).

Answer:

A. 2