Easier question?

Question

ABCD is any quadrilateral having each of its sides tangent to a circle of diameter 9 cm. If the perimeter of ABCD is 56 cm, what is the area of ABCD?

Answer 1

S=56*(9/2)/2=126

Answer 2

Nice question.

I strongly suspect (wouldn't want to prove) that the quadrilateral must be a rhombus. (A square obviously doesn't work, because it would have to have sides of length 9 and therefore a perimeter of 36.)

I hope it's ok to cheat a little by looking at similar problems. I found this one: http://mathcentral.uregina.ca/QQ/database/QQ.09.00/jacky4.html

In this case we have x (it's 9/2 = 4.5), and we have the hypotenuse (one side of the rhombus: 56/4 = 14), and we're looking for the other two sides of the triangle.

So the ratio of the length of the longer "half-diagonal" (lhd) to the hypotenuse (14) must be equal to the ratio of the length of the radius (4.5) to the length of the smaller half-diagonal (shd):

lhd/14 = 4.5/shd
lhd = 63/shd

And by the pythagorean theorem:

lhd^2 + shd^2 = 14^2 = 196
(3969/shd^2) + shd^2 = 196
shd^4 - 196shd^2 + 3969 = 0

Again I cheated a little and used a solver (URL below) to get the two answers of 4.78888 and 13.1555. So the length of the full diagonals would be 9.57776 and 26.311.

So the area, 1/2*d1*d2 = 252.

(I assume if I hadn't resorted to decimals, all the radicals would have disappeared at some point...since the answer came out to be 252.000044336...)

Answer 3

I'm taking a wild guess and saying 196. If it is a quadridical where all CORNERS are tangent to the circle, then it must be a square. If it is a square, then all sides are congruent. If all sides are congruent, then 56/4 would equal 14. The area would be 14*14, making 196.

Without more information, that's the best I can do.

Answer 4

I don't know the answer but definitely all of the 4 sides of a quadrilateral can be tangent to a circle.

Answer 5

Doesn't make sense:
"In geometry, a quadrilateral is a polygon with four sides and four vertices." You have a pentagon.

"In geometry, a pentagon is any five-sided polygon. However, the term is commonly used to mean a regular pentagon, where all sides are equal and all angles are equal (to 108°). Its Schläfli symbol is {5}.

The area of a regular pentagon with side length a is given by A = (5a^2/4)*(cotangent(pi/5)) = (a^2/4)*(sqrt(25+(10\sqrt(5))) = [approx] 1.72048*(a^2)"

From Wikipedia at:
http://en.wikipedia.org/wiki/Pentagon

Hope this helps!