<h3>Answer: 0.65</h3>
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Explanation:
Refer to the figure below.
Angle is 72 degrees because we divide a full 360 degree rotation into five equal pieces. This divides in half to get angle = 72/2 = 36 degrees.
Focus on triangle
Use the tangent ratio to find that...
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From that, determine the area of triangle BAK
There are five of these identical triangles to form the entire pentagon ABCDE. The triangles are a rotated copy of one another.
The area of pentagon ABCDE is 5*5.505524 = 27.52762 square units approximately.
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Now focus on right triangle LKF.
We'll use the cosine ratio to determine KL.
Staying with the same triangle, we'll use the sine ratio this time to find the length of LF
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Now find the area of triangle KGF
The area of the blue pentagon is roughly 5*3.603415 = 18.017075 square units.
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To summarize so far
- area of pentagon ABCDE (red) = 27.52762 square units approximately
- area of pentagon FGHIJ (blue) = 18.017075 square units approximately
We divide the two areas to get the ratio we want.
18.017075/27.52762 = 0.654509
Then this rounds to <u>0.65</u> when rounding to two decimal places.
The blue pentagon takes up roughly 65% of the area of the red pentagon.
In other words, if you threw a dart at random at the red figure, then you have around a 65% chance of landing inside the blue figure.