Question

select the correct answer.
the united states pentagon building is modeled on the coordinate plane as regular pentagon ( abcde ) with a scale ratio of 1 unit : 100 feet.
the vertices of the pentagon are ( a(-7.42, 2.42) ), ( b(0, 7.88) ), ( c(7.42, 2.42) ), ( d(4.605, -6.35) ), and ( e(-4.605, -6.35) ).
what is the approximate perimeter, in feet, of the u.s. pentagon building?

• ( 46.05 ) feet
• ( 6345 ) feet
• ( 63.45 ) feet
• ( 4605 ) feet

Explanation:

Step1: Identify the type of pentagon

The pentagon is regular, so all sides are equal. We can find the length of one side and multiply by 5.

Step2: Calculate the length of side \( AB \) (or any side, since it's regular)

Using the distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \): \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

For points \( A(-7.42, 2.42) \) and \( B(0, 7.88) \):
\( x_1 = -7.42, y_1 = 2.42 \); \( x_2 = 0, y_2 = 7.88 \)
\( d = \sqrt{(0 - (-7.42))^2 + (7.88 - 2.42)^2} \)
\( = \sqrt{(7.42)^2 + (5.46)^2} \)
\( = \sqrt{55.0564 + 29.8116} \)
\( = \sqrt{84.868} \approx 9.21 \) units

Step3: Find the length of side \( DE \) (horizontal side for simplicity)

Points \( D(4.605, -6.35) \) and \( E(-4.605, -6.35) \)
Since \( y \)-coordinates are the same, length is \( |x_2 - x_1| = | -4.605 - 4.605 | = 9.21 \) units (matches the regular pentagon side length)

Step4: Calculate perimeter in units

Perimeter of regular pentagon: \( 5 \times \) side length \( = 5 \times 9.21 = 46.05 \) units

Step5: Convert units to feet (scale ratio 1 unit : 100 feet)

Perimeter in feet: \( 46.05 \times 100 = 4605 \) feet? Wait, no—wait, wait, correction: Wait, earlier calculation for \( AB \) was wrong? Wait, no, let's recheck \( DE \):

Wait, \( D(4.605, -6.35) \) and \( E(-4.605, -6.35) \): the distance is \( 4.605 - (-4.605) = 9.21 \) units. Correct.

But wait, the answer options: 46.05, 6345, 63.45, 4605.

Wait, maybe I made a mistake in step 2. Let's recalculate \( AB \):

\( x_2 - x_1 = 0 - (-7.42) = 7.42 \)

\( y_2 - y_1 = 7.88 - 2.42 = 5.46 \)

\( 7.42^2 = 55.0564 \)

\( 5.46^2 = 29.8116 \)

Sum: \( 55.0564 + 29.8116 = 84.868 \)

Square root: \( \sqrt{84.868} \approx 9.21 \) units. So each side is ~9.21 units.

Perimeter in units: \( 5 \times 9.21 = 46.05 \) units.

Now, scale ratio: 1 unit = 100 feet. So perimeter in feet: \( 46.05 \times 100 = 4605 \) feet. Wait, but the option 4605 is there. But wait, let's check \( DE \) again: \( D(4.605, -6.35) \), \( E(-4.605, -6.35) \): distance is \( 4.605 - (-4.605) = 9.21 \) units. So each side is 9.21 units. Perimeter 5*9.21=46.05 units. Multiply by 100: 4605 feet.

Wait, but the answer option 4605 is present. So that's the answer.

Answer:

4605 feet (Option: 4605 feet)