Answer with explanation:
Given: In △ABC with side lengths a, b, and c, and height h, a vertical line extends from angle B perpendicular to side b.
To prove:→ Area(ΔABC)
[tex]=\frac{1}{2}\times ab \sin C[/tex]
Proof:
In △ABC with side lengths a, b, and c, and height h ,
Area of ΔABC
[tex]=\frac{1}{2}\times {\text{Base} \times {\text{height}[/tex]
Let perpendicular from angle , B on the side AC cuts AC at M.
Definition of sine
[tex]\sin {\text{theta}}=\frac{\text{Perpendicular}}{\text{Hypotenuse}}\\\\\sin C=\frac{BM}{BC}\\\\ \sin C=\frac{h}{a}\\\\h=a\sin C[/tex]
Area of Triangle ΔABC
[tex]=\frac{1}{2}\times b \times h\\\\=\frac{1}{2}\times b \times a \times \sin C[/tex]
-----Using Substitution Property,that is , h=a sin C
→Area = 1/2 b h
