Question

area and perimeter of triangles
deriving an area formula
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 height

If \(c\) is the base of \(\triangle ABC\), the height is the perpendicular distance from the opposite - vertex \(C\) to the line containing the base \(c\). In right - triangle \(BCD\), the height is \(CD\).

Step2: Use sine function

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

Step3: Calculate area

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

Answer:

If \(c\) is the base of \(\triangle ABC\), the height is \(CD\); \(\sin(A)=\frac{CD}{b}\); \(Area_{\triangle ABC}=\frac{1}{2}bc\sin(A)\)