5 is to 2…. as 100 is to what?

5………100

— = ———

2 …….. x

Since the numerator gets multiplied by 20 to get 100 (because 5 * 20 = 100)… you have to do the same to the denominator….

So … 2 * 20 = ?

2 * 20 = 40…. so the area of the smaller polygon is 40

PROBLEM 2:

Well, first we have to know what an apothem is…. Check out this link ( http://en.wikipedia.org/wiki/Apothem ) for the definition of an apothem.

Now the next two links are hints as to how to solve this problem…

http://mathcentral.uregina.ca/QQ/database/QQ.02.06/ester1.html

http://mathcentral.uregina.ca/QQ/database/QQ.02.06/student6.html

If you know the length of the apothem and the side length, then you can construct two identical right triangles for each side. That means 12 of these triangles in all. Since the apothem always bisects the side, you know the height and width of the triangles and can calculate the area, multiply by the number of triangles and get your answer.

So according to http://en.wikipedia.org/wiki/Apothem … the formula is…

A = nsa / 2 … where n = 12 (triangles), s = 2 sqrt 3 cm, and the apothem “a” = 3 cm…

A = [12 * (2 sqrt 3 cm) * 3 cm ] / 2 = 62.35 cm^2

PROBLEM 3:

You know that the area of a circle is A = pi * r^2…. so if your circle has an area of 16 pi…. that means that r^2 = 16… which automatically means that your radius is 4. So r = 4…

NOW…. you also know that the formula for circumference of a circle is C = 2 * pi * r…. so that means with a circle of radius 4 (we determined that r = 4 from the given area), the circumference of the cirlce is C = 2 * pi * 4 = 8 pi

PROBLEM 4:

Okay if the radius of the circle is 6… then that means that the DIAMETER of the circle is 6 * 2 = 12… right?

What you have to realize is that when you inscribe a square in the circle… and you draw a diagonal in the square… you get two right triangles… where the diameter of the circle also happens to be the hypoteneuse of these two right triangles. You then know that each of the base angles of the two right triangles are 45 degrees each…. because all the angles in a triangle equal to 180 degrees…. The 90 angles of the square… also happen to be the right angles of the right triangles…

So basically all you have to do is find the length of the sides of the triangle… which also happens to be the length of the side of the square…

So Pythagorea’s theorem states…

a^2 + b^2 = c^2

where a = b…. and c = 12 (the diameter of the circle)

… so then you have…

a^2 + a^2 = c^2

2a^2 = 12^2

2a^2 = 144

a^2 = 72…. which also happens to be the area of the square.

PROBLEM 5:

You know that an equilateral triangle has 3 sides that are the same lengths. So if your perimeter is 12… that means that each side is 4…. because 4 + 4 + 4 = 12, right?

Okay… now what is the formula for the area of a triangle?

A = (1/2) b * h

where b is the base of the triangle, and he is the height or “altitude” of the triangle (which we have to determine…

Well, to determine the height, we can use either pythagoreas’ theorem (a^2 + b^2 = c^2)… or we can use trigonometry….

I’m going to use trigonometry…. because I know SOH CAH TOA…. I also know that each angle is 60 degrees… I also know that when I draw a line straight from the peak of the triangle to the base… it bisects the base and it the line also creates two right triangles…

I know that the height of an equilateral triangle is OPPOSITE each of the base angles… I also know the hypoteneuse of these right triangles are the same as the sides of the equilateral triangle….

So… I’m going to use SOH… which stands for

Sin (Angle) = opposite leg / hypoteneuse

That means… opposite leg = Sin 60 degrees x hypoteneuse

Opposite leg = ht. of equilateral triangle = [(sqrt 3) / 2] x 4 = 2 sqrt 3

So… h = 2 sqrt 3

That means…. Area (of equilateral triangle) = (1/2) b * h

A = (1/2)(4)(2 sqrt 3) = 4 sqrt 3

2)K…I dont feel like working anymore its summer, why do you need these anyways?