Math Problem Statement
plz solve these manually woth answers
Solution
Let's work through each triangle step by step.
Problem (a):
We are given a triangle with:
- ,
- ,
- ,
- A right angle at , and
- The unknown side .
Using the sine rule in the right-angled triangle : Let's calculate .
Problem (b):
We are given a right-angled triangle with:
- ,
- ,
- ,
- (unknown angle), and
- Right angles at and .
We can use the tangent rule to find the unknown angle : Let's calculate .
Problem (c):
We have a circle and the triangle , where:
- ,
- ,
- , and
- (unknown).
Using the cosine rule: Let's calculate .
Problem (d):
We have:
- ,
- ,
- ,
- , and
- is the unknown side.
We can use the sine rule to find : Let's calculate .
Now, I will calculate all the unknowns.Here are the calculated results for each triangle:
(a)
The length of side (denoted ) is approximately 7.61 cm.
(b)
The unknown angle is approximately 18.4°.
(c)
The angle in triangle is approximately 93.8°.
(d)
The length of side (denoted ) is approximately 11.75 cm.
Would you like further details on these solutions, or have any specific questions?
Here are 5 related questions for further practice:
- How would you solve triangle (a) using the cosine rule instead?
- Can you use the sine rule to find the missing angles in triangle (b)?
- How would the solution change in problem (c) if one side length were altered?
- What is the general formula for solving triangles using the tangent rule?
- How can the sine rule be applied to solve non-right-angled triangles?
Tip: The sine and cosine rules are most useful when you have non-right-angled triangles or when you know two angles or two sides.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Sine Rule
Cosine Rule
Tangent Rule
Geometry of Circles
Formulas
Sine Rule: sin(A)/a = sin(B)/b = sin(C)/c
Cosine Rule: c^2 = a^2 + b^2 - 2ab * cos(C)
Tangent Rule: tan(θ) = opposite/adjacent
Theorems
Pythagorean Theorem
Tangent of an Angle
Cosine Rule for Non-Right-Angled Triangles
Sine Rule for Angles and Sides
Suitable Grade Level
Grades 9-12
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