MHB ASVAB circle and inscribed rectangle area problem

AI Thread Summary
The problem involves calculating the area of a shaded region formed by a rectangle inscribed in a circle. Given the rectangle's dimensions of 5 and 12, the diameter of the circle is determined to be 13, leading to a circle area of approximately 132.73. The area of the rectangle is calculated as 60, resulting in a shaded area of 72.7 when subtracting the rectangle's area from the circle's area. The discussion also touches on using coordinate geometry to confirm the calculations, emphasizing the relationship between the rectangle's sides and the circle's radius. The final answer for the area of the shaded region is confirmed to be option c, 72.7.
karush
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Rectangle ABCD is inscribed in the circle shown.
If the length of side $\overline{AB}$ is 5 and the length of side $\overline{BC}$ is 12
what is the area of the shaded region?

71.png


$a.\ 40.8\quad b.\ 53.1\quad c\ 72.7\quad d \ 78.5\quad e\ 81.7$

well to start with the common triangle of 12 5 13 gives us the diameter of 13
area of the circle is $\pi \left(\frac{13}{2}\right)^2=132.73$
area of {circle - retangle)=$132.73-60 =72.7$ which is c

typos maybe:unsure:
 
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assuming $\overline{BC} = 12$, not $\overline{AB}$

$\pi \left(\dfrac{d}{2}\right)^2 - |AB| \cdot |BC|$
 
I, because of my "prejudices", would immediately go to coordinate methods. Set up a Cartesian coordinate system with center at the center of the circle and with axes parallel to the sides of the rectangle.

The circle can be written $x^2+ y^2= R^2$. The line AB is x= -6 and CD is x= 6. The line BC is y= 5/2 and the line AD is y= -5/2.

The line y= 5/2 intersects the circle where $x^2+ 25/4= R^2$, so $x^2= R^2- 25/4$- that is at $\left(\sqrt{R^2- 25/4}, 5/2\right)$.
The line x= 6 intersects the circle where $36+ y^2= R^2$, so $y^2= R^2- 36$- that is at $\left(6, \sqrt{R^2- 36}\right)$.

Of course, those refer to the same point so we must have $6= \sqrt{R^2- 25/4}$ and $5/2= \sqrt{R^2- 36}$. The first is equivalent to $36= R^2- 25/4$ and the second to $25/4= R^2- 36$ so both give $R^2= 36+ 25/4= (144+ 25)/4= 169/4$ so $R= \pm\frac{13}{2}$. Since the radius of this circle is a positive number, $R= \frac{13}{2}$.

The circle has radius $\frac{13}{2}$ so area $\frac{169\pi}{4}$. The area of the rectangle is $12(5)= 60$ so the area of the shaded region is $\frac{169\pi}{4}- 60= \frac{169\pi- 240}{4}$.
 
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well this would be a lot easier if it wasn't decimal
 
skeeter said:
assuming $\overline{BC} = 12$, not $\overline{AB}$

$\pi \left(\dfrac{d}{2}\right)^2 - |AB| \cdot |BC|$
not sure if I have seen this notation before: |AB| as a distance between?
 
karush said:
not sure if I have seen this notation before: |AB| as a distance between?

yes
 
Jolly Good Mate!
 
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