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Re: If rectangle ABCD is inscribed in the circle above, what is the area
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26 Apr 2019, 10:05
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The rectangle is equally placed on the center of the circle. Let O be the center of the circle. And X be the midpoint of AB. Then OX = 5/2 = 2/5 and XB = 12/2 = 6 Triangle OXB is a right angles triangle. OB = radius = \(\sqrt{6^2+2.5^2} = \sqrt{42.5}\)
Draw a line connecting points A and C. An important circle property (see video below for more info) tells us that, if we have a 90-degree inscribed angle, then that angle must be containing ("holding") the DIAMETER of the circle. So, we know that AC = the diameter of the circle.
To find the hypotenuse of the red triangle, we can EITHER apply the Pythagorean Theorem OR recognize that 5 and 12 are part of the Pythagorean triplet 5-12-13 With either approach, we learn that AC = 13
IMPORTANT: if the diameter (AC) is 13, then the radius = 13/2 = 6.5
What is the area of the circular region? Area of circle = πr² So, area = π(6.5²) = 42.25π
If rectangle ABCD is inscribed in the circle above, what is the area
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27 Apr 2019, 16:57
Hi All,
We're told that a rectangle with sides of 5 and 12 is inscribed in the circle. We're asked for the area of the circular region. This question is based on a number of math patterns that can help you to save time answering the question.
First, any square or rectangle that is inscribed in a circle will have a diagonal that IS the diameter of the circle. Second, when splitting a rectangle or square across its diagonal, you'll form two right triangles. With the given side lengths of the rectangle (5 and 12), we have a 5/12/13 right triangle, so we know the diameter of the circle is 13. The radius is half the diameter... radius = 6.5
Area of a circle = (π)(R^2) = (π)(6.5^2)
At this point, you don't actually have to calculate the value. Since 6^2 = 36 and 7^2 = 49, we know the correct answer has to be between 36π and 49π. There's only one answer that fits...
We see that diagonal AC (of the rectangle) is also the diameter of the circle. When we make diagonal AC, we see that we create a right triangle, and more specifically a 5-12-13 right triangle, so the length of AC is 13.
Since the diameter is 13, the radius is 6.5, and the area of the circle is (6.5)^2 x π = 42.25π.
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68% (01:38) correct 
