Question

find the exact perimeter of hexagon abcdef plotted below.
b(1,5) c(6,5)
a(-6,2)
d(6,-1)
f(-6,-3) e(1,-3)
choose 1 answer:
a 34
b 52 + √29
c 23 + 3√29
d 23 + √29 + √58

Explanation:

Step1: Calculate length of AB

Use distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. For A(-6,2) and B(1,5):
$AB = \sqrt{(1 - (-6))^2 + (5 - 2)^2} = \sqrt{7^2 + 3^2} = \sqrt{49 + 9} = \sqrt{58}$

Step2: Calculate length of BC

Points B(1,5) and C(6,5). Since y-coordinates are same, distance is $|6 - 1| = 5$

Step3: Calculate length of CD

Points C(6,5) and D(6,-1). Since x-coordinates are same, distance is $| -1 - 5| = 6$

Step4: Calculate length of DE

Points D(6,-1) and E(1,-3). Use distance formula:
$DE = \sqrt{(1 - 6)^2 + (-3 - (-1))^2} = \sqrt{(-5)^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29}$

Step5: Calculate length of EF

Points E(1,-3) and F(-6,-3). Since y-coordinates are same, distance is $| -6 - 1| = 7$

Step6: Calculate length of FA

Points F(-6,-3) and A(-6,2). Since x-coordinates are same, distance is $|2 - (-3)| = 5$

Step7: Sum all lengths

Perimeter = $AB + BC + CD + DE + EF + FA$
= $\sqrt{58} + 5 + 6 + \sqrt{29} + 7 + 5$
= $23 + \sqrt{29} + \sqrt{58}$

Answer:

D. $23 + \sqrt{29} + \sqrt{58}$