Question

O secanta intersecteaza laturile unghiului XOY de masura 120° in A si B, iar bisectoarea lui in C. Prin C se duce paralela la AO care intersecteaza pe OY in D.

a) Stabiliti natura triunghiului OCD

b) Aratati ca 1/OC=1/OB+1/OA

[In care / reprezinta supra]


Va multumesc!

Answer 1

$\it a)\ \ CD||AO \Rightarrow m(\widehat{CDO}) = 180^0-m(\widehat{AOD})=180^0-120^0=60^0\ \ \ (1)\\ \\ OC - bisectoare \Rightarrow m(\widehat{COD})=120^0:2=60^0\ \ \ (2)\\ \\ Triunghiul\ OCD\ are\ dou\breve{a}\ unghiuri\ de\ 60^0 \Rightarrow \Delta OCD - echilateral$

$\it\ b)\ \ OC-bisectoare \Rightarrow OC=\dfrac{2\cdot OA\cdot OB}{OA+OB}cos60^0= \dfrac{2\cdot OA\cdot OB}{OA+OB}\cdot\dfrac{1}{2} \Rightarrow\\ \\ \\ \Rightarrow OC= \dfrac{OA\cdot OB}{OA+OB}\Rightarrow \dfrac{1}{OC}=\dfrac{OA+OB}{OA\cdot OB}\Rightarrow\dfrac{1}{OC}=\dfrac{OA}{OA\cdot OB}+\dfrac{OB}{OA\cdot OB}\Rightarrow\\ \\ \\ \Rightarrow \dfrac{1}{OC}=\dfrac{1}{OB}+\dfrac{1}{OA}.$