A triangle has two sides measuring 7.2 m and 8.5 m. The angle between these two sides is 46 degrees. Solve the triangle (find all missing sides and missing angles). Determine the perimeter and area of this triangle. Express all answers to nearest tenth explain in detail

Answer 1

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.