Question

1) Aflaţi măsurile unghiurilor unui triunghi ABC,ştiind că: a) sunt direct proporţionale cu numerele 7, 11 şi 12; b) sunt invers proporţionale cu numerele 0,2; 0,(3) şi 1; c) m( Â) = 3m(B) + 5° iar m(B) = m( Ĉ) – 25º.​

Answer 1

Răspuns:

Suma măsurilor unghiurilor unui triunghi este 180°.

a) a/7 = b/11 = c/12 = k

a = 7k

b = 11k

c = 12k

30k = 180

k = 6

a = 42°

b = 66°

c = 72°

b)

0,2 = 2/10 = 1/5

0,(3) = 3/9 = 1/3

a/5 = b/3 = c = k

a = 5k

b = 3k

c = k

9k = 180

k = 20

a = 100°

b = 60°

c = 20°

c) a = 3b + 5 => a = 3(c - 25) + 5 => a = 3c - 70

b = c - 25

3c - 70 + c - 25 + c = 180

5c = 275

c = 55°

a = 95°

b = 30°

Answer 2

$\it Not\breve am\ cu\ x,\ y,\ z\ m\breve asurile\ celor\ trei\ unghiuri.\\ \\ x+y+z=180^o\ \ \ \ (*)$

$\bf a)\ \it \{x,y,z\}\ d.p. \{7,\ 11,\ 12\}\ \Rightarrow \dfrac{x}{7}=\dfrac{y}{11}=\dfrac{z}{12}=\dfrac{x+y+z}{7+11+12}\ \stackrel{(*)}{=}\ \dfrac{180^o}{30}=6^o\\ \\ \\ \dfrac{x}{7}=6^o\Rightarrow x=7\cdot6^o \Rightarrow x=42^o\\ \\ \\ \dfrac{y}{11}=6^o\Rightarrow y=11\cdot6^o \Rightarrow y=66^o\\ \\ \\ \dfrac{z}{12}=6^o\Rightarrow z=12\cdot6^o \Rightarrow z=72^o$

$\bf b)\ \it 0,2=\dfrac{\ 2^{(2}}{10}=\dfrac{1}{5};\ \ 0,(3)=\dfrac{\ 3^{(3}}{9}=\dfrac{1}{3}\\ \\ \\ \{x,\ y,\ z\}\ i.\ p.\ \{\dfrac{1}{5},\ \dfrac{1}{3},\ 1\}\ \Rightarrow \{x,\ y,\ z\}\ d.\ p.\ \{5,\ 3,\ 1\}\ \Rightarrow \dfrac{x}{5}=\dfrac{y}{3}=\dfrac{z}{1}=\\ \\ \\ =\dfrac{x+y+z}{5+3+1}\ \stackrel{(*)}{=}\ \dfrac{180^o}{9}=20^o$

$\it \dfrac{x}{5}=20^o\Rightarrow x=5\cdot20^o \Rightarrow x=100^o\\ \\ \\ \dfrac{y}{3}=20^o\Rightarrow y=3\cdot20^o \Rightarrow y=60^o\\ \\ \\ \dfrac{z}{1}=20^o\ \Rightarrow z=20^o$