CONTOH SOAL 7 LINGKARAN (SEGI EMPAT TALI BUSUR)

 $\begin{aligned}31.\quad&\textrm{Perhatikan gambar berikut}\end{aligned}$.

$\begin{aligned}\quad\qquad&\textrm{Besar}\:\: \angle C+\angle D=\: .... \\ &\text{a}.\quad 145^{\displaystyle 0}\\ &\text{b}.\quad 180^{\displaystyle 0}\\ &\text{c}.\quad 215^{\displaystyle 0}\\ &\text{d}.\quad 290^{\displaystyle 0}\\\\ &\textrm{Jawab}:\quad \textbf{c}\\ &\begin{aligned} &\textrm{Perhatikan bahwa}\:\: \angle A+\angle B+\angle C+\angle D=360^{\displaystyle 0}\\ &\textrm{akibat dari segi empat tali busur. Sehingga besar}\\ &\angle A+\angle B+\angle C+\angle D=360^{\displaystyle 0}\\ &80^{\displaystyle 0}+65^{\displaystyle 0}+\angle C+\angle D=360^{\displaystyle 0}\\ &\angle C+\angle D=360^{\displaystyle 0}-(80^{\displaystyle 0}+65^{\displaystyle 0})=215^{\displaystyle 0}\end{aligned}\end{aligned}$.

$\begin{aligned}32.\quad&\textrm{Perhatikan gambar berikut}\end{aligned}$.

$\begin{aligned}\quad\qquad&\textrm{Jika}\:\: \angle PSO=3x^{\displaystyle 0},\: \angle PQO=2x^{\displaystyle 0},\:\: \textrm{serta}\:\:\: \angle PQRS=75^{\displaystyle 0}\\ &\textrm{maka nilai}\:\:x=\: .... \\ &\text{a}.\quad 15^{\displaystyle 0}\\ &\text{b}.\quad 21^{\displaystyle 0}\\ &\text{c}.\quad 45^{\displaystyle 0}\\ &\text{d}.\quad 50^{\displaystyle 0}\\\\ &\textrm{Jawab}:\quad \textbf{b}\\ &\begin{aligned}&\textrm{Perhatikan ilustrasi berikut} \end{aligned}\end{aligned}$.

$\begin{aligned}\quad\qquad&\angle SPQ=3x^{\displaystyle 0}+2x^{\displaystyle 0}=105^{\displaystyle 0}\\ &\textrm{Ingat bahwa}\:\:  \angle P +\angle R=180^{\displaystyle 0}\:\: \textrm{serta segitiga}\\ &\textrm{SPO dan QPO adalah segita sama kaki, sehingga}\\ &\angle SPQ=3x^{\displaystyle 0}+2x^{\displaystyle 0}=5x^{\displaystyle 0}=105^{\displaystyle 0}\\&\Leftrightarrow x^{\displaystyle 0}=21^{\displaystyle 0}\end{aligned}$.

$\begin{aligned}33.\quad&\textrm{Perhatikan gambar berikut}\end{aligned}$.

$\begin{aligned}\quad\qquad&\textrm{Jika}\:\: \angle QRT=108^{\displaystyle 0},\: \:\: \textrm{dan}\:\:\: PQ=PS\\ &\textrm{maka}\:\:\angle PQO=\: .... \\ &\text{a}.\quad 36^{\displaystyle 0}\\ &\text{b}.\quad 42^{\displaystyle 0}\\ &\text{c}.\quad 48^{\displaystyle 0}\\ &\text{d}.\quad 54^{\displaystyle 0}\\\\ &\textrm{Jawab}:\quad \textbf{d}\\ &\begin{aligned}&\textrm{Perhatikan ilustrasi berikut} \end{aligned}\end{aligned}$.

$\begin{aligned}\quad\qquad &\angle QRS=180^{\displaystyle 0}-\angle QRT\quad (\textrm{sudut pelurus})\\ &\:\qquad\quad=180^{\displaystyle 0}-108^{\displaystyle 0}=72^{\displaystyle 0}\\ &\angle QPS+\angle QRS=180^{\displaystyle 0}\Leftrightarrow \angle QPS=180^{\displaystyle 0}-\angle QRS\\ &(\textrm{akibat segi empat tali busur})\\&\bullet \quad\textrm{Bagian Pertama}:\bigtriangleup PSQ\\&\angle QPS=180^{\displaystyle 0}-72^{\displaystyle 0}=108^{\displaystyle 0}.\: \textrm{Karena}\: \bigtriangleup PSQ\: \textrm{sama kaki},\\ &\textrm{maka}\:\: \angle PQS=\angle PSQ.\:\: \textrm{Selanjutnya untuk}\:\: \bigtriangleup PSQ\\ &\angle PQS+\angle PSQ+\angle QPS=180^{\displaystyle 0}\\ &\Leftrightarrow 2\angle PQS+108^{\displaystyle 0}=180^{\displaystyle 0}\Leftrightarrow \angle PQS=36^{\displaystyle 0}\\ &\bullet \quad\textrm{Bagian Kedua}:\bigtriangleup SQO\\ &\angle QPS\:\: \textrm{dan}\:\: \angle QOS\:\: \textrm{menghadap busur besar yang}\\&\textrm{sama, yaitu}\:\:  \widehat{QS},\:\: \textrm{akibatnya}:\angle QOS=2\angle QPS=216^{\displaystyle 0}.\\ &\textrm{Akibat lanjutannya adalah}\\ &\angle QOS_{\displaystyle \textrm{busur kecil}}=360^{\displaystyle 0}-216^{\displaystyle 0}=144^{\displaystyle 0}.\\ &\textrm{Karena}\:\: \bigtriangleup SQO\:\: \textrm{sama kaki juga, maka}\:\: \angle OQS=\angle OSQ\\ &\textrm{Selanjutnya}:\angle OQS+\angle OSQ+\angle QOS=180^{\displaystyle 0}\\  & \Leftrightarrow 2\angle OQS+144^{\displaystyle 0}=180^{\displaystyle 0}\Leftrightarrow \angle OQS=18^{\displaystyle 0}.\\ &\textrm{Sehingga besar}\:\: \angle PQO=\angle PQS+\angle OQS\\ &\qquad\qquad\qquad\qquad\qquad=36^{\displaystyle 0}+18^{\displaystyle 0}=54^{\displaystyle 0}\end{aligned}$.

$\begin{aligned}34.\quad&\textrm{Perhatikan gambar berikut}\end{aligned}$.

$\begin{aligned}\quad\qquad&\textrm{Nilai}\:\:x-y=\: .... \\ &\text{a}.\quad 51^{\displaystyle 0}\\ &\text{b}.\quad 37^{\displaystyle 0}\\ &\text{c}.\quad 28^{\displaystyle 0}\\ &\text{d}.\quad 21^{\displaystyle 0}\\\\ &\textrm{Jawab}:\quad \textbf{b}\\ &\begin{aligned}&\textrm{Perhatikan bahwa}\: \begin{cases} 2x+3^{\displaystyle 0} &+&75^{\displaystyle 0} =180 \\ 3y-3^{\displaystyle 0} &+&140^{\displaystyle 0} =180 \end{cases}\\ &\textrm{Selanjutnya}\:\: \begin{cases} 2x & =180-75^{\displaystyle 0}-3^{\displaystyle 0}&\Leftrightarrow x=51^{\displaystyle 0} \\ 3y& =180-140^{\displaystyle 0}+3^{\displaystyle 0}&\Leftrightarrow y=14,\overline{33}^{\displaystyle 0} \end{cases}\\ &\textrm{Sehingga nilai}:x-y=51^{\displaystyle 0}-14,\overline{33}^{\displaystyle 0}=36,67^{\displaystyle 0}\simeq  37^{\displaystyle 0}\end{aligned}\end{aligned}$.

$\begin{aligned}35.\quad&\textrm{Perhatikan gambar berikut}\end{aligned}$.

$\begin{aligned}\quad\qquad&\textrm{Jika}\:\: \angle P:\angle Q:\angle R=5:8:7,\: \:\: \textrm{maka}\:\:\angle R=\: .... \\ &\text{a}.\quad 112^{\displaystyle 0}\\ &\text{b}.\quad 105^{\displaystyle 0}\\ &\text{c}.\quad 75^{\displaystyle 0}\\ &\text{d}.\quad 56^{\displaystyle 0}\\\\ &\textrm{Jawab}:\quad \textbf{b}\\ &\textrm{misalkan}:\angle P=5n,\:\angle Q=8n,\: \textrm{dan}\:\: \angle R=7n\\ &\textrm{Selanjutnya}\\ &\angle P+\angle Q=180^{\displaystyle 0}\Leftrightarrow 5n+7n=180^{\displaystyle 0}\Leftrightarrow 12n=180^{\displaystyle 0}\\&\textrm{sehingga}\quad n=15^{\displaystyle 0}\:\: \textrm{dan besar}\:\: \angle R=7n=7\times 15^{\displaystyle 0}=105^{\displaystyle 0}\end{aligned}$.