To solve this triangle, we can first use the Law of Cosines to find the third side:
c^2 = a^2 + b^2 - 2ab * cos(C)
Where c is the unknown side, a and b are the given sides, and C is the angle between the given sides.
c^2 = 7.2^2 + 8.5^2 - 2 * 7.2 * 8.5 * cos(46)
c^2 = 51.84 + 72.25 - 122.4 * cos(46)
c^2 = 51.84 + 72.25 - 78.308
c^2 = 45.782
c ≈ √45.782
c ≈ 6.8m
Now we can use the Law of Sines to find one of the missing angles:
sin(A) / a = sin(B) / b = sin(C) / c
sin(A) / 7.2 = sin(46) / 6.8
sin(A) = (7.2 * sin(46)) / 6.8
sin(A) ≈ 0.862
A ≈ sin^(-1)(0.862)
A ≈ 59.4 degrees
Now we can find the last angle:
C = 180 - 46 - 59.4
C ≈ 74.6 degrees
Now we have all the angles and sides of the triangle:
Angle A = 59.4 degrees
Angle B = 46 degrees
Angle C = 74.6 degrees
Side a = 7.2 m
Side b = 8.5 m
Side c ≈ 6.8 m
To find the perimeter, we add up all the sides:
Perimeter = 7.2 + 8.5 + 6.8
Perimeter ≈ 22.5m
To find the area, we can use Heron's formula:
s = (a + b + c) / 2
s = (7.2 + 8.5 + 6.8) / 2
s = 22.5 / 2
s = 11.25
Area = √(s(s-a)(s-b)(s-c))
Area = √(11.25(11.25-7.2)(11.25-8.5)(11.25-6.8))
Area = √(11.25 * 4.05 * 2.75 * 4.45)
Area ≈ 20.45 sq. meters
Therefore, the missing sides and angles of the triangle are:
Angle A = 59.4 degrees
Angle B = 46 degrees
Angle C = 74.6 degrees
Side a = 7.2 m
Side b = 8.5 m
Side c ≈ 6.8 m
The perimeter of the triangle is approximately 22.5 meters and the area is approximately 20.45 square meters.