Question

Cho \(\Delta ABC\left(AB>AC\right)\) vuông tại A. \(AB+AC+BC=4\); \(AB=BC-1\). Tính chu vi \(\Delta\)

Answer 1

AB+AC+BC=4

=>BC-1+AC+BC=4

=>2BC+AC=5

=>AC=5-2BC

AB>AC

nên BC-1>5-2BC

=>3BC>6

=>BC>2

Ta có: \(AB^2+AC^2=BC^2\)

\(\Leftrightarrow\left(BC-1\right)^2+\left(-2BC+5\right)^2=BC^2\)

\(\Leftrightarrow BC^2-2BC+1+4BC^2-20BC+25=BC^2\)

\(\Leftrightarrow4BC^2-22BC+26=0\)

\(\Leftrightarrow BC=\dfrac{11-\sqrt{17}}{4}\)

\(\Leftrightarrow AB=\dfrac{7-\sqrt{17}}{4};AC=\dfrac{\sqrt{17}-1}{2}\)

\(C=\dfrac{11-\sqrt{17}+7-\sqrt{17}+2\sqrt{17}-2}{4}=\dfrac{16}{4}=4\)