Question

the sides of $\triangle abc$ are 2, 3, and 4. which set of numbers could represent the sides of a triangle similar to $\triangle abc$?\
\\(\circ\\) \\{5,6,7\\}\
\\(\circ\\) \\{20,30,40\\}\
\\(\circ\\) \\{6,9,16\\}\
\\(\circ\\) \\{12,13,14\\}\
\
5 multiple choice 4 points\
the accompanying diagram shows two similar triangles.\
diagram of two similar triangles, first triangle has sides 24, 32, x; second triangle has sides 9, 12, 15\
which proportion could be used to solve for $x$?\
\\(\circ\\) $\frac{32}{12} = \frac{15}{x}$\
\\(\circ\\) $\frac{32}{x} = \frac{12}{15}$\
\\(\circ\\) $\frac{x}{24} = \frac{9}{15}$\
\\(\circ\\) $\frac{24}{9} = \frac{15}{x}$

Explanation:

First Question

Step1: Find similarity ratio condition

Similar triangles have proportional sides.

Step2: Check each option

• For $\{5,6,7\}$: $\frac{5}{2}

eq \frac{6}{3}
eq \frac{7}{4}$

• For $\{20,30,40\}$: $\frac{20}{2}=10$, $\frac{30}{3}=10$, $\frac{40}{4}=10$ (all equal)
• For $\{6,9,16\}$: $\frac{6}{2}

eq \frac{9}{3}
eq \frac{16}{4}$

• For $\{12,13,14\}$: $\frac{12}{2}

eq \frac{13}{3}
eq \frac{14}{4}$

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Second Question

Step1: Identify corresponding sides

In similar triangles, the side of length 24 corresponds to 9, side $x$ corresponds to 15, and side 32 corresponds to 12.

Step2: Set up correct proportion

Corresponding sides are proportional: $\frac{\text{Side of large triangle}}{\text{Side of small triangle}} = \frac{24}{9} = \frac{x}{15}$
This matches the structure of the option $\frac{24}{9} = \frac{15}{x}$ (likely a formatting error in the option; the correct proportion uses $x$ in the numerator of the right-hand side).

Answer:

$\{20, 30, 40\}$