Questions: Determine the number of triangles ABC possible with the given parts. A=37.4°, a=3.8, c=17.4 How many possible solutions does this triangle have?

Determine the number of triangles ABC possible with the given parts. A=37.4°, a=3.8, c=17.4 How many possible solutions does this triangle have?
Transcript text: Determine the number of triangles ABC possible with the given parts. \[ \mathrm{A}=37.4^{\circ} \quad \mathrm{a}=3.8 \quad \mathrm{c}=17.4 \] How many possible solutions does this triangle have? $\square$
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Solution

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Solution Steps

Step 1: Calculate the Height

Using the given angle \( A = 37.4^\circ \) and side \( c = 17.4 \), we convert the angle to radians: \[ A_{\text{rad}} = 0.6528 \] Next, we calculate the height \( h \) of the triangle using the formula: \[ h = c \cdot \sin(A_{\text{rad}}) = 17.4 \cdot \sin(0.6528) \approx 10.5683 \]

Step 2: Compare the Side Length

We compare the length of side \( a = 3.8 \) with the calculated height \( h \):

  • Since \( a < h \) (i.e., \( 3.8 < 10.5683 \)), this indicates that there are no possible triangles that can be formed with the given parts.

Final Answer

\(\boxed{0}\)

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