Answer:
Given : In △ABC, m∠A=60°, m∠C=45°,and AB=8 unit
Firstly, find the angles B
Sum of measures of the three angles of any triangle equal to the straight angle, and also expressed as 180 degree
∴m∠A+ m∠B+m∠C=180 ......[1]
Substitute the values of m∠A=60° and m∠C=45° in [1]
![60^{\circ}+ m\angle B+45^{\circ}=180^{\circ}](https://tex.z-dn.net/?f=60%5E%7B%5Ccirc%7D%2B%20m%5Cangle%20B%2B45%5E%7B%5Ccirc%7D%3D180%5E%7B%5Ccirc%7D)
![105^{\circ}+ m\angle B=180^{\circ}](https://tex.z-dn.net/?f=105%5E%7B%5Ccirc%7D%2B%20m%5Cangle%20B%3D180%5E%7B%5Ccirc%7D)
Simplify:
![m\angle B=75^{\circ}](https://tex.z-dn.net/?f=m%5Cangle%20B%3D75%5E%7B%5Ccirc%7D)
Now, find the sides of BC
For this, we can use law of sines,
Law of sine rule is an equation relating the lengths of the sides of a triangle to the sines of its angles.
![\frac{\sin A}{BC} = \frac{\sin C}{AB}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csin%20A%7D%7BBC%7D%20%3D%20%5Cfrac%7B%5Csin%20C%7D%7BAB%7D)
Substitute the values of ∠A=60°, ∠C=45°,and AB=8 unit to find BC.
![\frac{\sin 60^{\circ}}{BC} =\frac{\sin 45^{\circ}}{8}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csin%2060%5E%7B%5Ccirc%7D%7D%7BBC%7D%20%3D%5Cfrac%7B%5Csin%2045%5E%7B%5Ccirc%7D%7D%7B8%7D)
then,
![BC = 8 \cdot \frac{\sin 60^{\circ}}{\sin 45^{\circ}}](https://tex.z-dn.net/?f=BC%20%3D%208%20%5Ccdot%20%5Cfrac%7B%5Csin%2060%5E%7B%5Ccirc%7D%7D%7B%5Csin%2045%5E%7B%5Ccirc%7D%7D)
unit
Similarly for AC:
![\frac{\sin B}{AC} = \frac{\sin C}{AB}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csin%20B%7D%7BAC%7D%20%3D%20%5Cfrac%7B%5Csin%20C%7D%7BAB%7D)
Substitute the values of ∠B=75°, ∠C=45°,and AB=8 unit to find AC.
![\frac{\sin 75^{\circ}}{AC} =\frac{\sin 45^{\circ}}{8}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csin%2075%5E%7B%5Ccirc%7D%7D%7BAC%7D%20%3D%5Cfrac%7B%5Csin%2045%5E%7B%5Ccirc%7D%7D%7B8%7D)
then,
![AC = 8 \cdot \frac{\sin 75^{\circ}}{\sin 45^{\circ}}](https://tex.z-dn.net/?f=AC%20%3D%208%20%5Ccdot%20%5Cfrac%7B%5Csin%2075%5E%7B%5Ccirc%7D%7D%7B%5Csin%2045%5E%7B%5Ccirc%7D%7D)
unit
To find the perimeter of triangle ABC;
Perimeter = Sum of the sides of a triangle
i,e
Perimeter of △ABC = AB+BC+AC = 8 +9.798+10.9283 = 28.726 unit.
To find the area(A) of triangle ABC ;
Use the formula:
![A = \frac{1}{2} \times AB \times AC \times \sin A](https://tex.z-dn.net/?f=A%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%5Ctimes%20AB%20%5Ctimes%20AC%20%5Ctimes%20%5Csin%20A)
Substitute the values in above formula to get area;
![A=\frac{1}{2} \times 8 \times 10.9283 \times \sin 60^{\circ}](https://tex.z-dn.net/?f=A%3D%5Cfrac%7B1%7D%7B2%7D%20%5Ctimes%208%20%5Ctimes%2010.9283%20%5Ctimes%20%5Csin%2060%5E%7B%5Ccirc%7D)
Simplify:
Area of triangle ABC = 37.856 (approx) square unit