Math Problem Statement

Is this a ssa situation

Solution

Yes, this is indeed an SSA (Side-Side-Angle) situation, which can lead to the ambiguous case in triangle determination. Let's carefully re-evaluate the SSA scenario for the given triangle:

Given:

  • A=58\angle A = 58^\circ
  • a=10.5a = 10.5 cm (side opposite A\angle A)
  • b=11.4b = 11.4 cm (another side of the triangle)

Steps to Determine the Number of Possible Triangles

  1. Calculate the Height of the Triangle:

    • Using the given angle and side aa, we calculate the height hh from vertex CC perpendicular to side ABAB. h=asin(A)h = a \cdot \sin(A) h=10.5sin(58)10.50.8488.904 cmh = 10.5 \cdot \sin(58^\circ) \approx 10.5 \cdot 0.848 \approx 8.904 \text{ cm}
  2. Compare the Height with Side bb:

    • b=11.4b = 11.4 cm
    • Since b>hb > h, we need to further analyze the comparison between bb and aa.
  3. Analyze the Lengths of Sides aa and bb:

    • If bab \le a, there could be two possible triangles.
    • If b>ab > a, there is exactly one possible triangle.

Evaluation:

  • Given h8.904h \approx 8.904 cm and b=11.4b = 11.4 cm, we note that b>hb > h.
  • Now, comparing bb with aa: b(11.4 cm)>a(10.5 cm)b (11.4 \text{ cm}) > a (10.5 \text{ cm})

Since b>ab > a, only one triangle can be formed.

Detailed Steps:

  1. Height Calculation: h=10.5sin(58)8.904 cmh = 10.5 \cdot \sin(58^\circ) \approx 8.904 \text{ cm}

  2. Comparison of bb and hh:

    • b=11.4b = 11.4 cm, which is greater than h=8.904h = 8.904 cm.
    • Since b>hb > h, there is a possibility for a triangle.
  3. Final Comparison of bb and aa:

    • b(11.4 cm)>a(10.5 cm)b (11.4 \text{ cm}) > a (10.5 \text{ cm}).
    • This confirms that there is exactly one possible triangle.

Conclusion:

Exactly one triangle can be formed with the given measurements.

Would you like further details or have any questions about this solution? Here are some follow-up questions you might consider:

  1. How do we use the Law of Sines in an SSA scenario?
  2. What are the specific conditions for zero, one, or two triangles in SSA cases?
  3. How can we visualize the ambiguous case with geometric drawings?
  4. How does the Law of Cosines apply differently from the Law of Sines?
  5. What are some examples of ambiguous cases in real-life applications?

Tip: When dealing with SSA cases, always calculate the height and compare it with the given sides to determine the possible number of triangles accurately.

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Math Problem Analysis

Mathematical Concepts

Triangle Geometry
Ambiguous Case in Triangles
Law of Sines

Formulas

Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)

Theorems

Ambiguous Case in SSA Triangles

Suitable Grade Level

High School