Question

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

Explanation:

Step1: Identify the height

The height of a triangle with base \(c\) (side \(AB\)) is the perpendicular distance from the opposite - vertex \(C\) to the line containing the base. So, if \(c\) is the base of \(\triangle ABC\), the height is \(CD\).

Step2: Define sine in right - triangle

In right - triangle \(ACD\), \(\sin(A)=\frac{CD}{b}\).

Step3: Derive the area formula

The area of a triangle is \(A = \frac{1}{2}\times base\times height\). Here, base \(= c\) and height \(= CD\). Since \(\sin(A)=\frac{CD}{b}\), then \(CD = b\sin(A)\). So, \(Area_{\triangle ABC}=\frac{1}{2}bc\sin(A)\).

Answer:

If \(c\) is the base of \(\triangle ABC\), the height is \(CD\); \(\sin(A)=\frac{CD}{b}\); The previous statement is leading to the derivation of the area formula \(Area_{\triangle ABC}=\frac{1}{2}bc\sin(A)\)