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: Recall similarity criteria

Similar triangles have proportional side - lengths and equal corresponding angles.

Step2: Analyze triangle \(ABC\) (with vertices \(A(2,2)\), \(B(8,8)\), \(C(14,2)\))

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

Step3: Analyze triangle \(DEF\) (with vertices \(D(16,5)\), \(E(20,9)\), \(F(24,5)\))

Base \(DF=24 - 16 = 8\), height (vertical distance from \(E\) to \(DF\)) \(=9 - 5=4\). The ratio of base to height for \(ABC\) is \(\frac{12}{6}=2\), and for \(DEF\) is \(\frac{8}{4}=2\).

Step4: Analyze triangle \(PQR\) (with vertices \(P(12,11)\), \(Q(20,17)\), \(R(28,11)\))

Base \(PR = 28 - 12=16\), height (vertical distance from \(Q\) to \(PR\)) \(=17 - 11 = 6\). The ratio of base to height for \(PQR\) is \(\frac{16}{6}=\frac{8}{3}
eq2\).

Step5: Analyze triangle \(GHI\) (with vertices \(G(0,20)\), \(H(16,20)\), \(I(8,12)\))

Base \(GH=16 - 0 = 16\), height (vertical distance from \(I\) to \(GH\)) \(=20 - 12 = 8\). The ratio of base to height for \(GHI\) is \(\frac{16}{8}=2\).

Since triangles \(ABC\), \(DEF\), and \(GHI\) have the same ratio of base to height (which implies similar triangles as the angles will be equal due to the slope - based relationships in coordinate geometry), there are \(3\) triangles that can be mapped to one another by similarity transformations.

Answer:

D. 3