Math Problem Statement
Solution
This problem involves a right-angled triangle where the angle , the length of side cm, and the hypotenuse cm. We need to find the remaining side and the other angles of the triangle.
Solution:
Given:
- cm (one of the legs)
- cm (the hypotenuse)
Using the Pythagorean theorem: Plugging in the known values: Subtract 9 from both sides: Now, take the square root of both sides:
Finding the Angles:
Since it's a right-angled triangle, we already have . We now need to find the other two angles, and .
Using trigonometry, we can find these angles. Let's start with :
Since the sum of angles in a triangle is :
Final Answer:
- The length of side cm.
- .
- .
Would you like more details or have any questions?
Further Questions:
- How would you solve this problem using the cosine or tangent functions?
- How can the sine rule be applied to find the angles of this triangle?
- What happens to the other angles if is increased to 4 cm?
- How does the Pythagorean theorem change if the triangle is not right-angled?
- Can you use the properties of special right triangles to simplify this problem?
Tip:
In right-angled triangles, knowing two sides or one side and one angle (other than the right angle) allows you to find all other sides and angles using basic trigonometric functions.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Right-Angled Triangles
Pythagorean Theorem
Formulas
AC² = AB² + BC² (Pythagorean theorem)
sin(θ) = opposite / hypotenuse
cos(θ) = adjacent / hypotenuse
tan(θ) = opposite / adjacent
Theorems
Pythagorean Theorem
Basic Trigonometric Functions
Suitable Grade Level
Grades 7-10
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