Question

fie triunghiul ABC dreptunghic in A, AD perpendicular pe BC
a) demonstrati relatia 1/AD^2=1/AB^2+1/AC^2
b) aratati ca sin^2 B+cos^2 B=1
c)daca masura unghiului B este de 30 de grade si mediana AM are lungimea de 10 cm calculati lungimea catetei AB

am nevoie urgent va roog

Answer 1

a)

$\it Folosim\ formula \ AD=\dfrac{AB\cdot AC}{BC} \Leftrightarrow AD^2=\dfrac{AB^2\cdot AC^2}{BC^2} \Leftrightarrow \\ \\ \\ \Leftrightarrow \dfrac{1}{AD^2} = \dfrac{BC^2}{AB^2\cdot AC^2} \ \stackrel{T.Pitagora}{\Longleftrightarrow}\ \dfrac{1}{AD^2} = \dfrac{AB^2+AC^2}{AB^2\cdot AC^2} \Leftrightarrow \\ \\ \\ \Leftrightarrow \dfrac{1}{AD^2} = \dfrac{AB^2}{AB^2\cdot AC^2} +\dfrac{AC^2}{AB^2\cdot AC^2} =\dfrac{1}{AC^2} +\dfrac{1}{AB^2} =\dfrac{1}{AB^2} +\dfrac{1}{AC^2}$

b)

$\it sin^2B+cos^2B= \dfrac{AC^2}{BC^2} +\dfrac{AB^2}{BC^2} =\dfrac{AC^2+AB^2}{BC^2} \\ \\ \\ Teorema\ lui \ Pitagora \Rightarrow AC^2+AB^2=BC^2, \ deci: \\ \\ sin^2B+cos^2B = \dfrac{BC^2}{BC^2} =1$

c)

Mediana corespunzătoare ipotenuzei are lungimea egală cu jumătatea lungimii ipotenuzei. BC/2 = AM=10 ⇒ BC = 2·10 =20 cm

Cu teorema unghiului de 30° ⇒ AC = BC/2=20/2 = 10cm

Cu teorema lui Pitagora ⇒ AB² = BC² - AC² = 20² - 10² =400-100=300 ⇒

⇒ AB = √300=√100·3= 10√3 cm