Math Problem Statement
Solution
The triangle provided has the following properties:
- Side
- Side
- Side
- Angle
We can solve for the unknowns in this triangle using the Law of Cosines, which is useful for non-right triangles.
Law of Cosines:
Where , , and are the sides of the triangle, and is the included angle between sides and .
For this triangle:
- (side )
- (side )
- (side )
Substitute into the Law of Cosines formula: Since , the equation simplifies to:
Now, solve for . Let me calculate that.The solutions for are:
Since represents a length in the triangle, only the positive solution is valid:
Thus, .
Would you like further details or have any questions?
Here are some follow-up questions to deepen your understanding:
- What is the value of the side now that we know ?
- How can the Law of Sines be applied to check the consistency of this solution?
- How would this problem change if the angle was different from ?
- Can you compute the area of the triangle using Heron's formula with these side lengths?
- What other types of triangles can we apply the Law of Cosines to?
Tip: The Law of Cosines is especially useful when you know two sides and the included angle, or when you know all three sides but need to find an angle.
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Math Problem Analysis
Mathematical Concepts
Geometry
Trigonometry
Formulas
Law of Cosines: c^2 = a^2 + b^2 - 2ab * cos(C)
Theorems
Law of Cosines
Suitable Grade Level
Grades 9-11