In a triangle with angles α,β,γ and opposite sides a, b, c you are given:
α=35°, β=100°, b=40.
Find all missing lengths and angles as well as the area correct to 1 decimal place.
The missing angle γ = 45°
by the sine law:
a/sin35 = 40/sin100
a = 40sin35/sin100 = appr 23.3
c/sin45 = 40/sin100
c = 40sin45/sin100 = appr 28.7
area = (1/2)(ab)sin45
= appr 329.5 units^2
To find the missing lengths and angles in the triangle, we can use the properties and formulas of triangles.
First, let's find the missing angle γ. The sum of the angles in a triangle is always 180°, so we can calculate γ using the equation:
γ = 180° - α - β
Substituting the given values, we have:
γ = 180° - 35° - 100°
γ = 45°
So the missing angle γ is 45°.
Next, let's find the missing side lengths using the Law of Sines. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant. In equation form, it is given as:
a/sinα = b/sinβ = c/sinγ
Let's find side a:
a/sinα = b/sinβ
a/sin(35°) = 40/sin(100°)
To find a, we can cross-multiply and solve for a:
a = (40 * sin(35°)) / sin(100°)
a ≈ 21.4 (rounded to 1 decimal place)
So the missing side length a is approximately 21.4.
Now, let's find side c:
a/sinα = c/sinγ
21.4/sin(35°) = c/sin(45°)
To find c, we can cross-multiply and solve for c:
c = (21.4 * sin(45°)) / sin(35°)
c ≈ 22.6 (rounded to 1 decimal place)
So the missing side length c is approximately 22.6.
Lastly, let's find the area of the triangle using the formula for the area of a triangle:
Area = (1/2) * base * height
Since we have side b (40) and its opposite angle α (35°), we can use the formula:
Area = (1/2) * b * a * sin(α)
Substituting the values, we have:
Area = (1/2) * 40 * 21.4 * sin(35°)
Area ≈ 294.3 (rounded to 1 decimal place)
So the area of the triangle is approximately 294.3 square units.