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IrinaK [193]
2 years ago
7

Find the area and perimeter of a nonagon, inscribed in a circle, with radius 4 inches​

Mathematics
1 answer:
Natasha2012 [34]2 years ago
7 0

If it's a regular nonagon, each exterior angle has measure 360°/9 = 40°, so the supplementary interior angles each have measure 180° - 40° = 140°.

We can split up the nonagon into 9 congruent isosceles triangles whose longer side coincides with the radius of the circle. (The language in the question is a bit ambiguous; I think it's clearler to say the nonagon is itself circumscribed by the circle, so that each of its vertices lie on the circle. See attached image - not mine, but it's in public domain.)

As the image suggests, each triangle will have interior angles measuring 70°, 70°, and 40°.

We can use trigonometry to find the side length of the nonagon, i.e. the length of the shortest side of each triangle - I'll call it x. For example, using the law of cosines,

x² = 4² + 4² - 2 • 4 • 4 cos(40°)   ⇒   x = 4√2 √(1 - cos(40°))

Then the <u>perimeter</u> of the nonagon is

9x = 36√2 √(1 - cos(40°)) in ≈ 24.623 in

The area of a triangle is 1/2 the product of its base and height. The base will be b = x, but to get the height we cut the isosceles triangle in half to produce a right triangle. We use trig again to determine its height h:

tan(70°) = h/(b/2)   ⇒   h = 1/2 x tan(70°)

so each triangle contributes an area of

a = 1/2 bh = 1/4 x² tan(70°) ≈ 5.142 in²

and so the <u>area</u> of the nonagon is

9a = 9x²/8 tan(70°) ≈ 46.281 in²

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6. Can an isosceles triangle be an obtuse triangle? Explain.
kondor19780726 [428]

Answer:

No. It can be obtuse angle are angles greater than 90° but less than 180°

And for an isosceles triangle, two angles must be equal so if two angles are greater than 90° that mean the sum of angle in a triangle is above 180° which is not possible

Step-by-step explanation:

5 0
3 years ago
Determine if the following scenario is inductive or deductive reasoning:
Elan Coil [88]
You guys should go eat and then come pick ya guys up and then go pick ya up and go pick ya up some stuff and go get some stuff to go pick up some lunch wyeywywywyywywywyeh was your day so you can come pick ya up and then come pick me up I wanna is the day I wanna come to hang out with you guys tomorrow night or tomorrow or Saturday or tomorrow night or tomorrow or Saturday or Sunday or Saturday or tomorrow or Saturday or Sunday or Saturday night or Sunday or 22
8 0
3 years ago
Justine ran for 35 minutes on monday, for 1 hour and 10 minutes on Wednesday , and for 1 hour and 15 minutes on friday. How many
Oduvanchick [21]

Answer:

3 hours or 180 minutes

Step-by-step explanation:

This one should be an easy one

First, lets convert everything to units

35 + (1 hour and 10 minutes or 70 minutes) + (1 hour and 15 minutes or 75 minutes) = 35 + 70 + 75 = 180 minutes, or 3 hours.

Hope it helps :)

7 0
2 years ago
PLEASE HELP!!! 15 PTS!!
kherson [118]

Answer:

\large\boxed{C.\ (2,\ -45^o)}

Step-by-step explanation:

Regular coordinates (x, y)

Polar coordinates (r, φ)

r=\sqrt{x^2+y^2}\\\\\psi=\arctan\frac{y}{x}

We have the point

(\sqrt2,\ -\sqrt2)

Substitute:

r=\sqrt{(\sqrt2)^2+(-\sqrt2)^2}=\sqrt{2+2}=\sqrt4=2\\\\\psi=\arctan\left(\frac{-\sqrt2}{\sqrt2}\right)=\arctan(-1)=-45^o

6 0
3 years ago
June bought three books. For the first book, she paid one half of her allowance plus $1 more. For the second book, she paid one
inysia [295]

Answer:

$34

Step-by-step explanation:

I'd work backwards.

For the third book, she paid all her remaining money. The problem says she paid "1/2 her leftover money + $3". This means that: (let m = money used to buy book 3)

m = 1/2m + 3

1/2m = 3

m = 6

For the second book: (let n = money before book 2)

n - m (money left after book 2) = 1/2n + 2

1/2n +2 is money used up for book, which is the same as n-m.

n = 1/2n + 2 + m

1/2n = 2 + m = 2 + 6

1/2n =8

n = 16

For the first book, she spent 1/2 her money + 1. If o = money before book 1 (or the whole allowance):

o - n = 1/2o + 1

o = 1/2o + n + 1 = 1/2o + 16 + 1

1/2o = 17

o = 34

Check!

Spent 17 (half 34) + 1 on book 1

16 left

Spent 8 (half 16) +2 on book 2

6 left

Spent 3 (half 6) + 3 on book 3

0 left

4 0
3 years ago
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