Geometry help...

[hamnet]

The following is part of a quilt design. The shaded part is formed by five identical 3 units by 3 units squares. Calculate the length of the diameter of the circle.

The picture: http://warcount.googlepages.com/square.JPG

The formula for diameter is d = 2√Area/pi. --> area of a circle is a = pi(r)^2. Would the radius be 3, 4.5, or 6?

Also: The height of a triangle is 2 m more than the length of its base. If the area of the triangle is 17.5 m^2, what is the length of its base?

17.5 = b(b+2)/2
17.5 = b^2 + 2b/2
35 = b^2 +2b
0 = b^2 +2b - 35
0 = (x+7)(x-5)
x=-7, x=5; however, when one of these numbers is placed in "b", shouldn't the equation be equal on both sides? If it is, perhaps I did something wrong?

 

[stapel]

Hint: One of the circle's diameters goes from the upper-left corner of the western square to the lower-right corner of the eastern square. That is, one diameter of the circle corresponds to a diagonal of the three-by-nine rectangle.

Eliz.

 

[soroban]

Hello, hamnet!

There is nothing wrong with your work . . .

The height of a triangle is 2 m more than the length of its base.
If the area of the triangle is 17.5 m², what is the length of its base?

\(\displaystyle \frac{b(b\,+\,2)}{2}\: =\: 17.5\;\;\Rightarrow\;\;\frac{b^2\,+\,2b}{2}\:=\:17.5\;\;\Rightarrow\;\;b^2\,+\,2b \:=\:35\)

\(\displaystyle b^2\,+\,2b\,-\,35\:=\:0\;\;\Rightarrow\;\;(b\,+\,7)(b\,-\,5)\:=\:0\;\;\Rightarrow\;\;b\,=\,-7,\;b\,=\,5\)

Of course, we reject the negative answer.

The length of the base is: \(\displaystyle \,\fbox{b\,=\,5\text{ m}}\)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

But both answers check out . . .

\(\displaystyle x\,=\,5:\;\;\frac{5(5+2)}{2}\:=\:\frac{(5)(7)}{2}\:=\:\frac{35}{2}\:=\:17.5\)

\(\displaystyle x\,=\,-7:\;\;\frac{(-7)(-7+2)}{2}\:=\:\frac{(-7)(-5)}{2}\:=\:\frac{35}{2}\:=\:17.5\)

 

[hamnet]

So the base would be 5 and the height would be 7, as we are adding 2 more to the base.

~~~~~~~~~~~~~~~`

Also for the first question, about the circle:

To find the area I would do the following:

I know that there are five identical 3 x 9 squares, going vertical and horizontal. If I were to draw a line from the top-western corner to the lower-eastern corner I would have made a diameter, and I think I would do the same for the other (right-northern corner, to left-southern corner). Anyway,

Area = pi(r)^2
Area = pi(4.5)^2
Area = pi(20.25)
Area = 63.61725124

D = 2√Area/Pi
D= 2√63.61725124/Pi
D= 9

Would this be correct? Or.... Would you use:

c^2 = a^2 + b^2
where c is the unknown, a is equal to 9, and be is equal to 3
c^2 = 90
√c^2 = √90
c = 9.486832981