SOLUTION: All the vertices of a rhombus lie on a circle. Find the area of the rhombus, if area of the circle is 1256 cm2. (Use π = 3.14)

Algebra ->  Parallelograms -> SOLUTION: All the vertices of a rhombus lie on a circle. Find the area of the rhombus, if area of the circle is 1256 cm2. (Use π = 3.14)      Log On


   

Question 240235: All the vertices of a rhombus lie on a circle. Find the area of the rhombus, if area of the circle is 1256 cm2. (Use π = 3.14)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Since...
  1. Both pairs of opposite angles in parallelograms are congruent and a rhombus is a parallelogram (which has 4 congruent sides), the opposite angles of a rhombus are congruent.
  2. The vertices of the rhombus lie on the circle, the angles of the rhombus are inscribed angles.
  3. The opposite angles are congruent their arcs will be congruent.
  4. The arcs of the opposite angles make the entire circle, 360 degrees. (See the drawing below if you need help seeing this.)
  5. Two congruent arcs that add up to 360 must be 180 each.
  6. Inscribed angles whose arcs are 180 must be 90.
  7. A rhombus with 4 90 degree angles is called a square.

the rhombus described in this problem must be a square. So we want to find the area of of a square and the area of a square is A+=+s%5E2. This means we need to find the side of the square. So how do we find the side of the square when all we know is that the area of the circle is 1256? Here's a drawing that should help:

In the drawing, we have a 45-45-90 right triangle. And the radius of the circle is the hypotenuse of the right triangle and the sides of the right triangle are 1/2 the length of the sides of the square. If we know the radius of the circle then we can use that to find the side of the square:
  1. From the area of the circle, which we know, we can find the radius:
    Area of a circle = pi%2Ar%5E2.
    Substitute in for the area and pi:
    1256+=+3.14r%5E2
    Divide both sides by 3.14:
    400+=+r%5E2
    Find the square root of each side (ignoring the negative because a radius is never negative):
    20+=+r
  2. Use the Pythagorean Theorem (or the properties of 45-45-90 right triangles) to find the legs of the triangle. Let x = a leg of the triangle. Then, according to the Pythagorean Theorem:
    x%5E2+%2B+x%5E2+=+20%5E2
    Solving this:
    2x%5E2+=+400
    x%5E2+=+200
    x+=+sqrt%28200%29 (Again ignoring the negative solution)
    x+=+10sqrt%282%29
  3. Since the side of the triangle, x, is 1/2 the side of the square, the side of the square will be 20sqrt%282%29

Now that we know the side of the square we can find the area of the square:
A+=+%2820sqrt%282%29%29%5E2
A+=+800