Question

Cho tg ABC vuong tai A co dg cao AH goc B=60do , HB=a.Tren canh AC lay diem K( K ok trung vs A va C) . Goi D la hung chieu cua A tren BK

a) Tinh do dai cac canh cua tg ABC theo a

b) C/m rang: SBHD =1/4SBKC . cos^2gocABD

Answer 1

a: Xét ΔHBA vuông tại H có cos B=HB/AB

nên AB=2a

=>BC=4a

\(AC=\sqrt{\left(4a\right)^2-\left(2a\right)^2}=2a\sqrt{3}\)

b: \(cos^2ABD=\left(\dfrac{BA}{BD}\right)^2=\dfrac{BA^2}{BD^2}=\dfrac{BH\cdot BC}{BD\cdot BD}=\dfrac{BK}{BC}\cdot\dfrac{BC}{BD}=\dfrac{BK}{BD}\)

BH*BC=BA^2

BD*BK=BA^2

DO đó: BH*BC=BD*BK

=>BH/BK=BD/BC

=>ΔBHD đồng dạng với ΔBKC

=>\(\dfrac{S_{BHD}}{S_{BKC}}=\left(\dfrac{BH}{BK}\right)^2=\dfrac{BH^2}{BK^2}=\dfrac{1}{4}\cdot cos^2ABD\)