Question

see the figure of \\( \triangle abc \\) with auxiliary lines added. if \\( c \\) is the base of \\( \triangle abc \\), the height is. \\( \sin ( a ) = \\). the previous statement is leading to the derivation of which area formula? area \\( \triangle abc = \\)

Explanation:

Step1: Recall the definition of height in a triangle

In a triangle, when \(c\) is the base, the height is the perpendicular distance from the opposite vertex to the base. Here, in \(\triangle ABC\) with base \(c\) (side \(AB\)), the height is the length of the perpendicular segment from \(C\) to the line containing \(AB\). If we consider the right - triangle formed (e.g., \(\triangle ACD\)), and using the side - length notation:
In right - triangle \(ACD\) (right - angled at \(D\)), \(\sin(A)=\frac{\text{opposite}}{\text{hypotenuse}}\). The opposite side to angle \(A\) is the height \(h\) (the length of \(CD\)) and the hypotenuse is \(b\) (side \(AC\)). So, \(\sin(A)=\frac{h}{b}\), which implies \(h = b\sin(A)\)

Step2: Recall the area formula of a triangle

The area formula of a triangle is \(A=\frac{1}{2}\times\text{base}\times\text{height}\). We know the base is \(c\) and the height \(h = b\sin(A)\)
Substitute \(h\) into the area formula: \(A=\frac{1}{2}bc\sin(A)\)

Answer:

The height is \(b\sin(A)\), \(\sin(A)=\frac{\text{height}}{b}\), and the area formula is \(\frac{1}{2}bc\sin(A)\)