Question

Cho\(\Delta ABC\) nhọn,BC=a,CA=b,AB=c.

CMR:

\(S_{\Delta ABC}=\frac{1}{2}bc.sinA=\frac{1}{2}ca.sinB=\frac{1}{2}ab.sinC\)

Answer 1

Kẻ 3 đg cao AD,BE,CF của ΔABC

+ \(\left\{{}\begin{matrix}sinA=\frac{BE}{c}\\sinB=\frac{CF}{a}\\sinC=\frac{AD}{b}\end{matrix}\right.\)

+ \(S_{ABC}=\frac{1}{2}\cdot BE\cdot b=\frac{1}{2}\cdot CF\cdot c=\frac{1}{2}\cdot AD\cdot a\)

\(\Rightarrow S_{ABC}=\frac{1}{2}bc\cdot\frac{BE}{c}=\frac{1}{2}ca\cdot\frac{CF}{a}=\frac{1}{2}ab\cdot\frac{AD}{b}\)

\(\Rightarrow S_{ABC}=\frac{1}{2}bc\cdot sinA=\frac{1}{2}ca\cdot sinB=\frac{1}{2}ab\cdot sinC\)