Question

Get brainliest if you answer correctly! Make sure show your work!

Answer 1

Answer: 0.65

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Explanation:

Refer to the figure below.

Angle $\text{A}\text{K}\text{B}$ is 72 degrees because we divide a full 360 degree rotation into five equal pieces. This divides in half to get angle $\text{F}\text{K}\text{B}$ = 72/2 = 36 degrees.

Focus on triangle $\text{F}\text{K}\text{B}$

Use the tangent ratio to find that...

$\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}\\\\\tan(\text{K}) = \frac{\text{FB}}{\text{KF}}\\\\\tan(36) = \frac{2}{\text{KF}}\\\\0.726543 \approx \frac{2}{\text{KF}}\\\\ \text{KF}\approx \frac{2}{0.726543}\\\\ \text{KF}\approx 2.752762$

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From that, determine the area of triangle BAK

$\text{area} = \frac{\text{base}*\text{height}}{2}\\\\\text{area} = \frac{\text{AB}*\text{KF}}{2}\\\\\text{area} \approx \frac{4*2.752762}{2}\\\\\text{area} \approx 5.505524$

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.

$\cos(\text{angle}) = \frac{\text{adjacent}}{\text{hypotenuse}}\\\\\cos(\text{K}) = \frac{\text{KL}}{\text{KF}}\\\\\cos(36) \approx \frac{\text{KL}}{2.752764}\\\\ \text{KL}\approx 2.752764\cos(36)\\\\ \text{KL}\approx 2.227033$

Staying with the same triangle, we'll use the sine ratio this time to find the length of LF

$\sin(\text{angle}) = \frac{\text{opposite}}{\text{hypotenuse}}\\\\\sin(\text{K}) = \frac{\text{LF}}{\text{KF}}\\\\\sin(36) \approx \frac{\text{LF}}{2.752764}\\\\ \text{LF}\approx 2.752764\sin(36)\\\\ \text{LF}\approx 1.618034$

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Now find the area of triangle KGF

$\text{area} = \frac{\text{base}*\text{height}}{2}\\\\\text{area} = \frac{\text{FG}*\text{KL}}{2}\\\\\text{area} = \frac{2*\text{LF}*\text{KL}}{2}\\\\\text{area} \approx \frac{2*1.618034*2.227033}{2}\\\\\text{area} \approx 3.603415$

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 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.