will someone please help me with some Geometry??

- anonymous

will someone please help me with some Geometry??

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- anonymous

yes?

- anonymous

how would you find the perimeter and area of these two regular polygons??

##### 2 Attachments

- anonymous

ouch
ok so for the triangle, assuming is equilateral, the 8 should be 2/3rds of the height of the triangle, so the height is 12.
That means that the leg of the inscribed triangle (shorter one) is 4 (the other 1/3rd of the absolute height of the overall triangle.) so you can use pythagoras theorem to find the side of the inscribed triangle (outer side that is ON the overall triangle) which should be exactly half of the side that it is drawn on.
that will give you the two numbers you need, where perimeter is 3 times the length per side aka 3*sqrt(48)
and the area is half the height times length which is (1/2)(12)(sqrt(48))

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- anonymous

so 6.93?

- anonymous

you gotta multiply it out, 6.93 is the length of that tiny inscribed line that goes to the center (not the one labeled 8, the other one) XD

- anonymous

i dont understand

- anonymous

so i used the properties of a equilateral triangle which states that the circumcenter or whatever that center is called, is precisely 2/3s its total distance from the outer angle, and 1/3 from the sides. which means that the line labeled 8, is 2/3 of the total length from any angle, directly to the side opposite of it

- anonymous

knowing that, we can also see that the shorter inscribed line (the one NOT labeled 8) is going to be the other 1/3rd of the total distance.
since 8 is 2/3rds of the total distance, we know the total distance from any angle to its opposite side, is 12 (8 is 2/3rds so 4 is the other 1/3 and 8+4 is 12)
hence we know that the OTHER inscribed angle (NOT labeled 8), because it is also a direct line from the center of the side to the opposite angle, is 4.
we can use Pythagoras theorem to find the dimensions of that inscribed triangle.
8 being the hyppoteneus, and 4 being one leg, the third leg is the sqare root of 8^2 - 4^2
which gives us the square root of 48. which is the length of the third side of the inscribed triangle. we can assume that the third side (the one that is on top of the overall triangles perimeter) is one half of that entire side of the overall triangle.
so that means that the sides of the overall triangle are each 2* sqrt48
which means the overall perimeter is 3 times 2 times the sqare root of 48. because the perimeter is the sum of the three sides added which are each 2 times the sqareroot of 48.
(3)(2)(sqrt48)
(3)(2)(6.93)
(6)(6.93)
(41.58)=perimeter
the area of any triangle is exactly one half of the absolute height times the length. we have our absolute height as 12, and we know our sides length as 2 times the root of 48
so the area is exactly one half of 12 times 2 times the root of 48
written as: (1/2)(12)(2)(sqrt48)
which can be instantly simplified since the one half and two cancel out to 1
(12)(sqrt 48) is our area
(12)(6.93) simplified
(83.16) is our fully simplified area

- anonymous

ok

- anonymous

are you allowed to use the trig functions on these?

- anonymous

idk

- anonymous

...have you learned sine cosing and tangent?

- anonymous

yea

- anonymous

ok well tangent gives us the opposite over adjacent

- anonymous

so in the case of the pentagon, we are given just barely enough info to figure it out, with tangent of course

- anonymous

we need to use the angles of the pentagon itself, so lets imagine that the line labeled 12 is making a triangle with one of the angles of the pentagon

- anonymous

ok

- anonymous

we know that 12 is the bigger leg, and that the smaller leg is half of the length of the side, and the hypoteneus is the length from any angle to the center

- anonymous

so its a 30, 60, 90

- anonymous

make sense?

- anonymous

no it actually isint, and youll see why in a second...

- anonymous

yea

- anonymous

ok

- anonymous

its actually a 36, 54, 90 and heres why:

- anonymous

lemmy draw a picture...

- anonymous

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- anonymous

so the red angle has to be 180 and the orange angle is the angle we want to use.
we know that all regular polygons have a way of finding that orange angle and that is by knowledge that the interior angles of a pentagon add up to 540. from there we know that each angle is 540/5. which means each angle is 108

- anonymous

ok

- anonymous

##### 1 Attachment

- anonymous

here i have drawn the triangle we will use, we know that the green angle is half of the pentagon angle, aka half of 108, which is 54

- anonymous

and a triangles angles add up to 180, so 90+54 is 144, leaving us with 36 for the top angle.

- anonymous

ok

- anonymous

so now we can use our tangent euation. lets label the triangle though, so i can communicate better, the bottom side will be called A, it is the shortest side of this triangle, the other leg, which is 12, is B, and the hypotenues is C.
the angle at the top, opposite of side A, is "a" and the 54 degree angle is "b"

- anonymous

so our tangent equation states that, the opposite of an angle divided by the adjacent is the same as tangent of the angle.

- anonymous

so lets use tangent on "a"
that means that A/C = tan(a)

- anonymous

ok

- anonymous

so:
A/12 = tan(36)
we use our calculator for tangent of 36
which gives us
A/12=.726542528
so we multiply both sides by 12 and we get
A=8.718510336

- anonymous

and remember, A is only half of the length of the side, so the whole side is two times A
2*8.718510336
is 17.43702067

- anonymous

so your perimeter is five times that, because we have five sides

- anonymous

as for the area, i am not sure about the actual equation but an easy way to find the area is to find the area of the triangle that we just used, and multiply it by 10, because we can fit 10 of those triangles in that pentagon... makes sense?

- anonymous

so p=87.19

- anonymous

yeah thats the perimeter

- anonymous

than what would be area?

- anonymous

remember what the area of a triangle was?

- anonymous

no

- anonymous

A=8.718510336

- anonymous

i wrote it above, in the last problem, it is half of the absolute height of a triangle times its base

- anonymous

haha no, its not 8.7, its far from 8.7 XD

- anonymous

17.43702067

- anonymous

ok so we know the area of a triangle is half the height times the base, which means we can find the area of the triangle i had drawn inside the pentagon. since the base A 8.718510336 and our absolute height B is 12, we just multiply those together then divide it in half:
(1/2)(8.718510336)(12)
(1/2)(104.622124)
(52.31106202) is the area of our triangle inside the pentagon.
since ten of those triangles fit inside this pentagon, the pentagons area is ten times the triangles or:
(52.31106202)(10)=
523.1106202 is our pentagons area

- anonymous

okay

- anonymous

so i hope you understood all that, goodluck with your geometry

- anonymous

thank you.. can you help me with a couple more

- anonymous

errrrr.... ill try

- anonymous

how would you find the ratio (larger to smaller) of their perimeters and their areas??

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- anonymous

if you are given the measure of each interior angle of a regular n-gon. Find the value of n. 1)108 degrees 2)157.5 degrees.. would n=16? 3)150 degrees 4)170 degrees

- anonymous

ok so for ratios, always remember that if two shapes are similar, their sides and ratios are simply given by: bigger side:smaller side, in that case:
2.5:2 and 7.5:6
or
2.5/2 ad 7.5/6
which we can check to see if they are true ratios if they equal the same thing and both equal 1.25 so those are your correct ratios for perimeter

- anonymous

ok

- anonymous

for areas, remember that area is just two dimensions, and is referenced as something units sqared.
it just so happens that rations also exhibit this, so the ratio of the lengths sqared is the ratio of the areas.
so (2.5/2)^2 is the ratio of our area
same with (7.5/6)^2
2.5^2 is 6.25 and 2^2 is 4
which gives us 6.25/4
which is 1.5625
same with the other one, 7.5 and 6 but i wont waste time doing that one, it should give you 1.5625 (which is, BTW, 1.25^2)

- anonymous

okay

- anonymous

so for the interior angles, the formulas are so:
angle times number of sides will give you the total angle
A*n=TA
total angle of any n-gon is 180(n-2)
TA=180(n-2)
so A*n=180(n-2)
so for 108 we set it up like so:
108n=180(n-2)
ditribute:
108n=180n-360
subtract 180n
-72n=-360
divide by -72
n=5
same with the other three problems

- anonymous

2)157.5 degrees.. would n=16?

- anonymous

yes, sorry for late reply

- anonymous

where did the 360 come from?

- anonymous

360 is the 180 times 2.
remember that we had 180(n-2)
so by distributing we multiply the 180 by both n AND the -2
so we get 180n and -360
or 180n-360

- anonymous

so it would be 150n=180(n-2)

- anonymous

yep

- anonymous

n=12

- anonymous

yep

- anonymous

170n=180(n-2)
n=36

- anonymous

exactly

- anonymous

ok thank you.. one last one please?

- anonymous

bring it

- anonymous

what would the probability be that a point chosen at random would lie in the shaded region?

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- anonymous

imagine if you could fold that drawing, and you folded the shaded regions into the square

- anonymous

ok

- anonymous

what would it look like? how much of the white square can you see?

- anonymous

what would it look like? how much of the white square can you see?

- anonymous

it would be folded onto a square and you wouldnt see any white

- anonymous

so this means that the shaded region is _____ to the square
fill in the blank: equal, greater than, or less than

- anonymous

equal

- anonymous

so 50%

- anonymous

exactly

- anonymous

thought technically, most people dont choose points near the outside haha but yeah the answer is 50%

- anonymous

thank you so much for your help

- anonymous

no problem

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