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]
[tex]60^{\circ}+ m\angle B+45^{\circ}=180^{\circ}[/tex]
[tex]105^{\circ}+ m\angle B=180^{\circ}[/tex]
Simplify:
[tex]m\angle B=75^{\circ}[/tex]
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.
[tex]\frac{\sin A}{BC} = \frac{\sin C}{AB}[/tex]
Substitute the values of ∠A=60°, ∠C=45°,and AB=8 unit to find BC.
[tex]\frac{\sin 60^{\circ}}{BC} =\frac{\sin 45^{\circ}}{8}[/tex]
then,
[tex]BC = 8 \cdot \frac{\sin 60^{\circ}}{\sin 45^{\circ}}[/tex]
[tex]BC=8 \cdot \frac{0.866025405}{0.707106781} =9.798[/tex] unit
Similarly for AC:
[tex]\frac{\sin B}{AC} = \frac{\sin C}{AB}[/tex]
Substitute the values of ∠B=75°, ∠C=45°,and AB=8 unit to find AC.
[tex]\frac{\sin 75^{\circ}}{AC} =\frac{\sin 45^{\circ}}{8}[/tex]
then,
[tex]AC = 8 \cdot \frac{\sin 75^{\circ}}{\sin 45^{\circ}}[/tex]
[tex]AC=8 \cdot \frac{0.96592582628}{0.707106781} =10.9283[/tex] 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:
[tex]A = \frac{1}{2} \times AB \times AC \times \sin A[/tex]
Substitute the values in above formula to get area;
[tex]A=\frac{1}{2} \times 8 \times 10.9283 \times \sin 60^{\circ}[/tex]
[tex]A = 4 \times 10.9283 \times 0.86602540378[/tex]
Simplify:
Area of triangle ABC = 37.856 (approx) square unit
