Окръжност, дъги, ъгли

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Здрасти, отново пиша, защото окръжностите ме затрудняват...мерси

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Без заглавие - 2022-05-20T155444.623.png (222.85 KiB) Прегледано 1059 пъти

[tex]\overset{\displaystyle\frown}{NM} : \overset{\displaystyle\frown}{MP} : \overset{\displaystyle\frown}{PQN} = 3 : 2 : 7 \Rightarrow \overset{\displaystyle\frown}{NM} = 3x , \overset{\displaystyle\frown}{MP} = 2x , \overset{\displaystyle\frown}{PQN} = 7x[/tex]
[tex]3x + 2x + 7x = 360 ^\circ \Leftrightarrow 12x = 360 ^\circ \Rightarrow x = 30 ^\circ[/tex]
[tex]\overset{\displaystyle\frown}{NM} = 3.30 ^\circ = 90 ^\circ , \overset{\displaystyle\frown}{MP} = 2.30 ^\circ = 60 ^\circ , \overset{\displaystyle\frown}{PQN} = 7.30 ^\circ = 210 ^\circ[/tex]

[tex]\angle CQN[/tex] е периферен [tex]\Rightarrow \angle CQN = \displaystyle \frac{\overset{\displaystyle\frown}{QN}}{2}[/tex]

[tex]\angle QBP = \displaystyle \frac{\overset{\displaystyle\frown}{QNMP} - \overset{\displaystyle\frown}{QP}}{2} = 80 ^\circ[/tex]
([tex]\angle QBP[/tex] е ъгъл между допирателните $BP$ и $BQ$ спуснати от външната точка $B$ към окръжността)

[tex]\overset {\displaystyle\frown}{QNMP} = \overset{\displaystyle\frown}{QN} + \overset{\displaystyle\frown}{NM} + \overset{\displaystyle\frown}{MP} = \overset{\displaystyle\frown}{QN} + 90 ^\circ + 60 ^\circ = \overset{\displaystyle\frown}{QN} + 150 ^\circ[/tex]
[tex]\overset{\displaystyle\frown}{QP} = 360 ^\circ - \overset{\displaystyle\frown}{QNMP} = 360 ^\circ - (\overset{\displaystyle\frown}{QN} + 150 ^\circ ) = 210 ^\circ - \overset{\displaystyle\frown}{QN}[/tex]
Тогава получавам за [tex]\angle QBP[/tex]:
[tex]\angle QBP = \displaystyle\frac{\overset{\displaystyle\frown}{QN} + 150 ^\circ - (210 ^\circ - \overset{\displaystyle\frown}{QN}) }{2} = 80 ^\circ \Leftrightarrow \displaystyle \frac{2.\overset{\displaystyle\frown}{QN} - 60 ^\circ }{2} = 80 ^\circ[/tex][tex]\Rightarrow \overset{\displaystyle\frown}{QN} = 110 ^\circ[/tex]
[tex]\angle CQN = \displaystyle \frac{\overset{\displaystyle\frown}{QN}}{2} = \displaystyle \frac{110 ^\circ }{2}[/tex]
$$\Rightarrow \angle CQN = 55 ^\circ $$