Question

. Find the exact circumference in terms of pi of the circles. (5 points each)

Answer 1

Answer:
`√(338)\pi`,
`25\pi`

Explanation:

I will start with the second one, since it will be easier. Because this is a right triangle, you can use the given side lengths of the triangle to find the hypotenuse, which is also equal to the diameter of the triangle. Since the pythagorean theorem states that a^2(the shorter leg, squared) + b^2 (the longer leg, squared) = c^2 (the hypotenuse, squared) we can turn it into an equation like this:

`7^(2\ )+\ 24^(2)\ =\ c^(2)`

(lets use x in place of c in the next few equations, for algebraic simplicity)

Simplify

`49+\ 576\ =x^(2)`

`625=x^(2)`

Get the square roots

`√(625)=\sqrt{x^(2)}`

`x=25`

Since x is equal to the hypotenuse, we know that the hypotenuse is also 25. This means the diameter of the second circle is 25, and we can now find the circumference. The EXACT circumference of the second circle is 25π, but a simplified version can be created by calculating it as if it were an equation. This comes out to around 78.5398163397, which would trail for infinity. which is why to get an exact measurement of the diameter, the most simplified exact version is displayed by multiplying the diameter by the pi symbol, because it is impossible to fit an infinite number in a finite space.

Now for the first circle, which I am now realising is no more difficult than the last one. We can do the same thing in this case, using pythagorean theorem, because there are two dash lines marking two of the lines on the circle to be equal to one another, we know that the two marked lines are equal to 13, despite not appearing to be.

We can use the same equation as we did before, but slightly modified.

`13^(2)+13^(2)=x^(2)`

Simplify:

`169+169=x^(2)`

`338=x^(2)`

Now we can take the square roots of both sides of the equation.

`√(338)=\sqrt{x^(2)}`

This is where we come to a bit of a pause. Because the square root of 338 is an irrational constant number, we have to use the unsimplified square root in our answer for the circumference, otherwise, our answer would not be exact.

The diameter of our circle is
`√(338)`, which means that we can use an equation like this to find the circumference:

`√(338)\pi`

But since we cannot simplify this any further without it not being exact, we must leave it as is.

Answer 2

Explanation:

The hypotenuse of each of these right triangles can be found using Pythagorean theorem....the hypotenuse is also the DIAMETER of the circle

diamter * pi = circumference

First one:

Hypot^2 = 13^2 + 13^2

hypot ^2 = 338

hypot = diameter = sqrt (338)

circumference = diamter * pi = sqrt 338 pi ft ( or 13 sqrt(2) pi ft )

The second one is the same with slightly different numbers...you try it ...