Problem
GEO-B2-M09-P001 A Coordinate Rotation
#1
★☆☆☆☆ Level 1 of 5
Let \(A(0,0)\), \(B(a,b)\), \(C(-b,a)\), where \((a,b)\ne (0,0)\). Prove that triangle \(ABC\) is right isosceles.
Compare \(AB^2\) and \(AC^2\), and check perpendicularity by the dot product.
\(\overrightarrow{AB}=(a,b)\), \(\overrightarrow{AC}=(-b,a)\). Then \(AB^2=a^2+b^2=AC^2\), and \(\overrightarrow{AB}\cdot\overrightarrow{AC}=a(-b)+ba=0\). Hence \(AB=AC\) and \(AB\perp AC\).
A good first exercise on the dot product.