Question

13. În patrulaterul convex ABCD, măsurile unghiurilor KA, KB, C şi KD sunt invers proporţionale cu numerele 2, 3, 4 şi 6. Aflaţi măsurile unghiurilor patrulaterului.​

Answer 1

Răspuns:

144°; 96°; 72°; 48°

Explicație pas cu pas:

patrulater convex:

$\measuredangle A + \measuredangle B + \measuredangle C + \measuredangle D = 360 \degree$

notăm:

$2\measuredangle A = 3\measuredangle B = 4\measuredangle C = 6\measuredangle D = k$

atunci:

$\measuredangle A = \dfrac{k}{2} \ ; \ \measuredangle B = \dfrac{k}{3} \ ; \ \measuredangle C = \dfrac{k}{4} \ ; \ \measuredangle D = \dfrac{k}{6}$

înlocuim:

$\dfrac{^{6)} k}{2} + \dfrac{^{4)}k}{3} + \dfrac{^{3)}k}{4} + \dfrac{^{2)}k}{6} = 360 \\ \dfrac{6k + 4k + 3k + 2k}{12} = 360 \\ 15k = 12 \times 360 \implies k = 288$

$\measuredangle A = \dfrac{288}{2} \implies \measuredangle A = 144 \degree \\ \measuredangle B = \dfrac{288}{3} \implies \measuredangle B = 96 \degree \\ \measuredangle C = \dfrac{288}{4} \implies \measuredangle C = 72 \degree \\ \measuredangle D = \dfrac{288}{6} \implies \measuredangle D = 48 \degree$