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Parts of a Circle Theory
![\begin{tabular}{p{12cm}c} \begin{tabular}{p{12cm}} A circle has centre \(C\).\newline \(A B\) is the \textbf{diameter} as it passes through \(C\).\newline \(CD\) is a \textbf{radius}, since it starts at \(C\) and ends on the circumference.\newline The straight line \(A D\) is a \textbf{chord} the curved line (part of the \newline circumference) is an arc. \end{tabular} & \begin{tabular}{c} \begin{tikzpicture}[scale=0.9,line width=1pt] \coordinate[label=above:C] (O) at (0,0); \coordinate[label=right:B] (B) at (right:2); \coordinate[label=left:A] (A) at (left:2); \coordinate[label=below right:D] (D) at (-30:2); \draw (O) circle (2cm); \draw (A)--(B); \draw (O)--(D); \draw (A)--(D); \fill (O) circle (2.5pt); \end{tikzpicture} \end{tabular} \end{tabular} \hfill \linebreak \begin{tabular}{p{12.5cm}c} \begin{tabular}{p{12cm}} A circle has centre \(C\)\newline The area \(ACB\) contained by the radii \(A C\) and \(B C\) and the minor arc \(A B\) is the minor sector.\newline The area \(ACB\) contained by the radii \(A C\) and \(B C\) and the major arc \(AB\) is the major sector.\newline The area contained by the chord DE and the minor arc \(DE\) is the minor segment.\newline The area contained by the chord \(D E\) and the major arc \(DE\) is the major segment. \end{tabular} & \begin{tabular}{c} \begin{tikzpicture}[scale=0.9,line width=1pt] \coordinate[label=above:C] (O) at (0,0); \coordinate[label=right:B] (B) at (-30:2); \coordinate[label=below:A] (A) at (250:2); \coordinate[label=above left:D] (D) at (130:2); \coordinate[label=above right:E] (E) at (50:2); \draw (O) circle (2cm); \draw (A)--(O)--(B); \draw (D)--(E); \fill (O) circle (2.5pt); \end{tikzpicture} \end{tabular} \end{tabular} \begin{multicols}{2} \textbf{Example 1}\\ A circle has a diameter of \(6 \text{~cm}\).\\ What is the length of \(CB\)?\\ \begin{center} \begin{tikzpicture}[scale=0.7,line width=1pt] \coordinate[label=above:C] (O) at (0,0); \coordinate[label=above left:B] (B) at (150:2); \coordinate[label=right:A] (A) at (-30:2); \draw (O) circle (2cm); \draw (A)--(B); \fill (O) circle (2.5pt); \end{tikzpicture} \end{center} \textbf{Example 1 solution}\\[3pt] \(\begin{aligned} CB & =\frac{1}{2} \times 6 \\ & =3 \text{~cm}\end{aligned}\) \columnbreak \textbf{Example 2}\\ The area of a circle is \(16 \pi \text{ cm}^2\) and its radius is \(4 \text{~cm}\).\\ \textbf{i)} what is the area of the minor sector \(ACB\)?\\ \textbf{ii)} What is the area of the minor segment contained by the chord \(A B\) and the minion arc \(A B\)?\\ \begin{center} \begin{tikzpicture}[scale=0.8,line width=1pt] \coordinate[label=above:C] (O) at (0,0); \coordinate[label=right:B] (B) at (right:2); \coordinate[label=below:A] (A) at (270:2); \draw[line width=1.5pt] (O) circle (2cm); \draw (A)--(O)--(B)--cycle; \fill (O) circle (2.5pt); \pic [draw=black,line width=1pt,angle radius=0.3cm,angle eccentricity=1.6,""] {right angle=B--O--A}; \end{tikzpicture} \end{center} \textbf{Example 2 solution}\\ \textbf{i)} The sector is one quarter of the circle\\ \(\begin{aligned} \therefore \text { Area } & =\frac{1}{4} \times 16 \pi =4 \pi \text{ cm}^2 \end{aligned}\)\\ \textbf{ii)} \(\begin{aligned}[t] \text{The area of the triangle} ACB&=\frac{1}{2} \times 4 \times 4=8 \text{~cm}^2 \end{aligned}\)\\ \(\therefore\) The area of the segment \(=(4 \pi-8) \text{ cm}^2\) \end{multicols}](/media/on0b2iep/7959.png)
NSW Y8 Maths Area and Perimeter Parts of a Circle
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