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Answer:
The area of a regular quadrilateral inscribed in a circle of radius 4 cm is [tex]32 \:cm^2[/tex].
Step-by-step explanation:
A quadrilateral is a polygon with four sides. So a regular quadrilateral is a shape that has four equal sides, with all the interior angles equal commonly known as a square.
To find the area of the square inscribed in a circle of radius 4 cm you must:
Step 1: From the graph below the diagonal of the square is 2r. We know that the radius is 4 cm. Therefore, the diagonal is 8 cm.
Step 2: We find the value of the sides with the help of the Pythagoras' Theorem.
Let a be a side of the square.
The Pythagorean Theorem tells us that the relationship in every right triangle is:
[tex]a^{2} +b^{2} =c^{2}[/tex]
For a square, the sides are equal (a = b, in the Pythagorean formula).
Therefore,
[tex]a^{2} +b^{2} =c^{2}\\a^{2} +a^{2} =c^{2}\\2a^{2} =c^{2}[/tex]
In our case c, represents the diagonal and we know that is equal to 8 cm.
The value of the sides is,
[tex]2a^{2} =8^{2}\\\\a^{2}=\frac{8^{2}}{2} \\\\a=\sqrt{\frac{8^{2}}{2} }[/tex]
Step 3: The area of the square is given by,
[tex]A=a^2[/tex]
Thus,
[tex]A=(\sqrt{\frac{8^{2}}{2} })^2\\\\A=\left(\left(\frac{8^2}{2}\right)^{\frac{1}{2}}\right)^2\\\\A=\frac{8^2}{2}=\frac{2^6}{2}=2^5=32 \:cm^2[/tex]
The area of a regular quadrilateral inscribed in a circle of radius 4 cm is;
32 cm²
A regular quadrilateral with 4 sides is simply a square. Therefore, if the square in inscribed in a circle with a radius of r = 4cm, it means the diagonal of the square will have a length that is twice its' radius.
Thus, length of diagonal = 2r = 2 × 4
Length of diagonal = 8 cm
Now, the sides of a square are equal. If the side of the square is x, then we can use Pythagoras theorem to get;
x² + x² = 8²
2x² = 64
x² = 64/2
x² = 32 cm²
Formula for are of the square is;
Area = x²
Thus, area = 32 cm²
In conclusion, the area of the square is 32 cm²
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