Construct a triangle abc, give the 2 sides ab 7.4 cm and ac 5.5 cm and an included angle at a 75 degrees. State the length of side bc

To find the length of side BC of triangle ABC , we can use the Law of Cosines, which states:
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
where c is the side opposite angle C .

Given that AB = 7.4 \, \text{cm} , AC = 5.5 \, \text{cm} , and angle A = 75^\circ , we can find angle C :
B = 180^\circ - A - C
C = 180^\circ - 75^\circ - \angle B

Now, we can plug in the known values into the Law of Cosines formula:
BC^2 = AB^2 + AC^2 - 2(AB)(AC) \cdot \cos(C)

After substituting the values and solving for BC , we get:
BC = \sqrt{AB^2 + AC^2 - 2(AB)(AC) \cdot \cos(C)}

BC = \sqrt{7.4^2 + 5.5^2 - 2(7.4)(5.5) \cdot \cos(105^\circ)}

BC = \sqrt{54.76 + 30.25 - 85.8 \cdot (-0.2588)}

BC = \sqrt{54.76 + 30.25 + 22.20948}

BC = \sqrt{107.21948}

\boxed{BC \approx 10.355 \, \text{cm}}