Resources for Arc Lengths Problems
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Arc Lengths Problems Theory
![\begin{multicols}{2} \textbf{Example 1}\\ Find the perimeter of the shape given that \(A B=4 \text{~cm}\) and \(BC=4 \text{~cm}\)\\ \begin{center} \begin{tikzpicture}[scale=0.7,line width=1pt] \coordinate[label=below:C] (O) at (0,0); \coordinate[label=below:B] (B) at (left:4); \coordinate[label=below:A] (A) at (left:8); \draw (O) arc (0:180:2); \draw (O) arc (0:180:4); \draw (A)--(B) node[below,midway] {4 cm}; \path (O)--(B) node[below,midway] {4 cm}; \end{tikzpicture} \end{center} \textbf{Example 1 solution}\\ \(\begin{aligned} \text{The semicircle on } AC:\; AC&=\frac{180^{\circ}}{360^{\circ}} \times 2 \pi \times 4\\ & =\frac{1}{2} \times 2 \pi \times 4 \\ & =4 \pi \text{ cm} \end{aligned}\)\\ \(\begin{aligned} \text{The semicircle on } B C:\;B C & =\frac{180^{\circ}}{360^{\circ}} \times 2 \pi \times 2 \\ & =\frac{1}{2} \times 2 \pi \times 2 \\ & =2 \pi \text{ cm} \end{aligned}\)\\ \(\begin{aligned} \therefore \text { Perimeter } & =4+4 \pi+2 \pi . \\ & =(4+6 \pi) \text{ cm}. \end{aligned}\) \columnbreak \textbf{Example 2}\\ Find the perimeter of the shape given that \(A C\) has a diameter of \(7 \text{~cm}\) The arc \(A B\) and the are \(B C\) have radii \(2.5 \text{~cm}\) and \(\theta=120^{\circ}\)\\ \begin{center} \begin{tikzpicture}[scale=0.4,line width=1pt] \coordinate[label=below:] (O) at (0,0); \coordinate[label=below:B] (O1) at (0,-2); \coordinate[label=right:C] (C) at (right:7); \coordinate[label=left:A] (A) at (left:7); \coordinate[label=below:E] (E) at (-4.3,-3.5); \coordinate[label=below:F] (F) at (4.5,-3.5); %\draw (E)--(O1) arc (20:140:2.5)--(E); %\draw (F)--(O1)+(right:4.35) arc (20:140:2)--(F); \draw (C) arc (0:180:7); \draw[dashed] (O)--(120:7) node[right,midway] {3.5 cm}; \draw[out=110,in=40,looseness=1] (O1) to (A); \draw[out=70,in=150,looseness=1] (O1) to (C); \draw[dashed] (A)--(E) node[left,midway]{2.5 cm}--(O1); \draw[dashed] (O1)--(F)--(C)node[right,midway]{2.5 cm}; \pic [draw=black,line width=1pt,angle radius=0.4cm,angle eccentricity=1.6,"\(120^{\circ}\)"] {angle=O1--E--A}; \pic [draw=black,line width=1pt,angle radius=0.4cm,angle eccentricity=1.6,"\(120^{\circ}\)"] {angle=C--F--O1}; \end{tikzpicture} \end{center} \textbf{Example 2 solution}\\ The semicircle \(A C:\) \; \(\begin{aligned}[t] A C&=\frac{180^{\circ}}{360^{\circ}} \times 2 \pi \times 3.5 \\ & =\frac{1}{2} \times 7 \pi \\ & =3.5 \pi \text{ cm} \end{aligned}\)\\ The arc \(A B:\)\; \(\begin{aligned}[t] AB&=\frac{120^{\circ}}{360^{\circ}} \times 2 \pi \times 2.5\\ & =\frac{1}{3} \times 5 \pi \\ & =\frac{5 \pi}{3} \end{aligned}\)\\ arc \(A B=\) arc \(B C\; \therefore B C=\dfrac{5 \pi}{3}\)\\ \(\begin{aligned} \therefore \text { Perimeter } & =3.5 \pi+\frac{5 \pi}{3}+\frac{5 \pi}{3} \\ & =\left(\frac{7}{2}+\frac{5}{3}+\frac{5}{3}\right) \pi=\frac{41 \pi}{6} \text{~cm} \end{aligned}\) \end{multicols}](/media/qatbadf5/7961.png)
NSW Y8 Maths Area and Perimeter Arc Lengths Problems
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Arc Lengths Problems - Video - Arc Lengths Problems 1
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Arc Lengths Problems - Video - Arc Lengths Problems 2
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