Answer: 0.65
============================================================
Explanation:
Refer to the figure below.
Angle [tex]\text{A}\text{K}\text{B}[/tex] is 72 degrees because we divide a full 360 degree rotation into five equal pieces. This divides in half to get angle [tex]\text{F}\text{K}\text{B}[/tex] = 72/2 = 36 degrees.
Focus on triangle [tex]\text{F}\text{K}\text{B}[/tex]
Use the tangent ratio to find that...
[tex]\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}\\\\\tan(\text{K}) = \frac{\text{FB}}{\text{KF}}\\\\\tan(36) = \frac{2}{\text{KF}}\\\\0.726543 \approx \frac{2}{\text{KF}}\\\\ \text{KF}\approx \frac{2}{0.726543}\\\\ \text{KF}\approx 2.752762\\\\[/tex]
--------------------
From that, determine the area of triangle BAK
[tex]\text{area} = \frac{\text{base}*\text{height}}{2}\\\\\text{area} = \frac{\text{AB}*\text{KF}}{2}\\\\\text{area} \approx \frac{4*2.752762}{2}\\\\\text{area} \approx 5.505524\\\\[/tex]
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.
--------------------
Now focus on right triangle LKF.
We'll use the cosine ratio to determine KL.
[tex]\cos(\text{angle}) = \frac{\text{adjacent}}{\text{hypotenuse}}\\\\\cos(\text{K}) = \frac{\text{KL}}{\text{KF}}\\\\\cos(36) \approx \frac{\text{KL}}{2.752764}\\\\ \text{KL}\approx 2.752764\cos(36)\\\\ \text{KL}\approx 2.227033\\\\[/tex]
Staying with the same triangle, we'll use the sine ratio this time to find the length of LF
[tex]\sin(\text{angle}) = \frac{\text{opposite}}{\text{hypotenuse}}\\\\\sin(\text{K}) = \frac{\text{LF}}{\text{KF}}\\\\\sin(36) \approx \frac{\text{LF}}{2.752764}\\\\ \text{LF}\approx 2.752764\sin(36)\\\\ \text{LF}\approx 1.618034\\\\[/tex]
--------------------
Now find the area of triangle KGF
[tex]\text{area} = \frac{\text{base}*\text{height}}{2}\\\\\text{area} = \frac{\text{FG}*\text{KL}}{2}\\\\\text{area} = \frac{2*\text{LF}*\text{KL}}{2}\\\\\text{area} \approx \frac{2*1.618034*2.227033}{2}\\\\\text{area} \approx 3.603415\\\\[/tex]
The area of the blue pentagon is roughly 5*3.603415 = 18.017075 square units.
--------------------
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 0.65 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.
