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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))
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
i dont understand
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
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
are you allowed to use the trig functions on these?
...have you learned sine cosing and tangent?
ok well tangent gives us the opposite over adjacent
so in the case of the pentagon, we are given just barely enough info to figure it out, with tangent of course
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
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
so its a 30, 60, 90
no it actually isint, and youll see why in a second...
its actually a 36, 54, 90 and heres why:
lemmy draw a picture...
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
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
and a triangles angles add up to 180, so 90+54 is 144, leaving us with 36 for the top angle.
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"
so our tangent equation states that, the opposite of an angle divided by the adjacent is the same as tangent of the angle.
so lets use tangent on "a" that means that A/C = tan(a)
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
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
so your perimeter is five times that, because we have five sides
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?
yeah thats the perimeter
than what would be area?
remember what the area of a triangle was?
i wrote it above, in the last problem, it is half of the absolute height of a triangle times its base
haha no, its not 8.7, its far from 8.7 XD
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
so i hope you understood all that, goodluck with your geometry
thank you.. can you help me with a couple more
errrrr.... ill try
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
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
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)
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
2)157.5 degrees.. would n=16?
yes, sorry for late reply
where did the 360 come from?
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
so it would be 150n=180(n-2)
ok thank you.. one last one please?
imagine if you could fold that drawing, and you folded the shaded regions into the square
what would it look like? how much of the white square can you see?
what would it look like? how much of the white square can you see?
it would be folded onto a square and you wouldnt see any white
so this means that the shaded region is _____ to the square fill in the blank: equal, greater than, or less than
thought technically, most people dont choose points near the outside haha but yeah the answer is 50%
thank you so much for your help