Math Problem Statement
Solution
The given figure shows a right triangle with a smaller right triangle embedded inside it. To solve for , we can use trigonometric relationships based on the angles provided.
Step 1: Label Analysis
- is a right triangle, where , , and .
- The triangle is also a right triangle, with and .
- The value represents the length , which is the side opposite the right angle in .
Step 2: Using Trigonometry in
In the smaller right triangle , we can apply the tangent function since we have the angle , and we know the length :
From trigonometry, we know that . Therefore, we can write:
Solving for :
Step 3: Applying the Pythagorean Theorem in
Now, in , we apply the Pythagorean theorem to find . We know that and .
However, since we already know the exact relationships from the geometry of the figure and basic trigonometry:
Let me know if you'd like more detailed steps or further clarifications!
Here are some follow-up questions for deeper understanding:
- What is the significance of the angle in the solution?
- How would the solution change if was instead of ?
- Can you use the sine function to solve for other sides of the triangle?
- How does the Pythagorean theorem apply in this problem?
- Could you solve for the area of using the dimensions provided?
Tip: When dealing with right triangles and angles, remember that tangent is a useful trigonometric function when you know the opposite and adjacent sides.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Pythagorean Theorem
Angle Properties in Triangles
Formulas
tan(60°) = opposite/adjacent
Pythagorean Theorem: a^2 + b^2 = c^2
Theorems
Pythagorean Theorem
Basic Trigonometric Ratios
Suitable Grade Level
Grades 9-11
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