Answer:
49.82 units
Step-by-step explanation:
The given composite figure is created by placing a triangle on a sector of a circle, where the radius of the sector of the circle is equal to the base of the triangle.
Assuming that the triangle is a right triangle, we can determine the length of its base by using the Pythagorean Theorem:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Pythagorean Theorem}}\\\\a^2+b^2=c^2\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$a$ and $b$ are the legs of the right triangle.}\\\phantom{ww}\bullet\;\textsf{$c$ is the hypotenuse (longest side) of the right triangle.}\\\end{array}}[/tex]
In this case, the hypotenuse is 15 units and the height is 9 units. Therefore, substitute c = 15 and a = 9 into the formula and solve for b:
[tex]9^2+b^2=15^2\\\\81+b^2=225\\\\b^2=144\\\\\sqrt{b^2}=\sqrt{144}\\\\b=12[/tex]
So, the base of the triangle is 12 units, which means that the radius of the sector of the circle is r = 12 units.
To find the arc length of the sector of the circle, we can use the arc length formula:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Arc length}}\\\\s= \pi r\left(\dfrac{\theta}{180^{\circ}}\right)\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$s$ is the arc length.}\\\phantom{ww}\bullet\;\textsf{$r$ is the radius.}\\\phantom{ww}\bullet\;\textsf{$\theta$ is the central angle in degrees.}\end{array}}[/tex]
Substitute r = 12, θ = 66° and π = 3.14 into the formula and solve for s:
[tex]s=3.14 \cdot 12\cdot \dfrac{66^{\circ}}{180^{\circ}}\\\\\\s=37.68\cdot 0.3666...\\\\\\s=13.816[/tex]
The perimeter of the composite figure is the sum of the two given sides of the triangle, the arc length, and one radius:
[tex]\textsf{Perimeter}=15+9+13.816+12\\\\\textsf{Perimeter}=49.816\\\\\textsf{Perimeter}=49.82\; \sf (nearest\;hundredth)[/tex]
Therefore, the perimeter of the composite figure is 49.82 units (rounded to the nearest hundredth).
