Math Problem Statement
DO NOT USE TRIG
Solution
Solution:
We are given the following:
- is a right triangle with ,
- ,
- ,
- .
We need to find:
- The length of ,
- The area of .
Step 1: Relationship Between Angles and Sides
Observations:
- Since is a right triangle and , is the hypotenuse.
- In , , and because the angles of a triangle sum to .
From the geometry of -- triangles:
- The side opposite (in this case ) is half the hypotenuse of (which is ).
- The side opposite (in this case ) is times the side opposite .
Applying the -- Triangle Rule:
Let . Then:
- ,
- .
Step 2: Relating and Using Pythagoras in
From : Substituting :
Step 3: Solve for Using
In , the full hypotenuse can be expressed as: From Step 1:
- ,
- (since ).
Thus:
Substitute into the Pythagorean equation:
Step 4: Expand and Simplify
Expand :
Equating this to :
Cancel from both sides:
Multiply through by 4 to eliminate the fraction:
Simplify:
Divide through by (since ):
Step 5: Find and the Area
-
-
Area of :
Final Answers:
- ,
- Area of
Do you have any questions or want further clarification?
Here are related questions to expand your understanding:
- How do the properties of -- triangles simplify solving geometric problems?
- What are the conditions for a perpendicular dropped from a vertex in a triangle?
- How does the Pythagorean Theorem apply in solving problems involving multiple triangles?
- What other methods could be used to find the area of a triangle besides base-height relationships?
- Can this solution be adapted for a triangle with different angles (not --)?
Tip: Always simplify triangle problems by analyzing smaller triangles within the given shape!
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Math Problem Analysis
Mathematical Concepts
Geometry
Right Triangles
Special Triangles
30-60-90 Triangle Properties
Formulas
Area of a triangle = (1/2) × base × height
Properties of 30-60-90 triangles
Theorems
Pythagorean Theorem
Special Right Triangle Theorem (30°-60°-90°)
Suitable Grade Level
Grade 9-11
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