SOLUTION: The circle with center O is inscribed in kite ABCD. Find the length of the radius of the circle, in cm. https://ibb.co/94Z8wrZ A) 9 {{{ 3/5 }}} B) 8 {{{ 5/11 }}} C) 11 {{{

Algebra ->  Circles -> SOLUTION: The circle with center O is inscribed in kite ABCD. Find the length of the radius of the circle, in cm. https://ibb.co/94Z8wrZ A) 9 {{{ 3/5 }}} B) 8 {{{ 5/11 }}} C) 11 {{{       Log On


   

Question 1199742: The circle with center O is inscribed in kite ABCD. Find the length of the radius of the circle, in cm.
https://ibb.co/94Z8wrZ
A) 9 +3%2F5+
B) 8 +5%2F11+
C) 11 +1%2F3+
D) 10 +2%2F7+
E) 12 +5%2F13+
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The radii OE and OF are perpendicular to the sides of the kite, so triangles AFO and OEC are both similar to triangle ADC.

Let r be the radius of the circle; let x be the length of segment FD.

In similar triangles AFO and ADC, AF/FO = AD/DC:

%2824-x%29%2Fr=24%2F18=4%2F3
4r=72-3x
3x%2B4r=72 [1]

In similar triangles OEC and ADC, EC/OE = DC/AD:

%2818-x%29%2Fr=18%2F24=3%2F4
3r=72-4x
4x%2B3r=72 [2]

Solve [1] and [2] for r by eliminating x.

12x%2B16r=288
12x%2B9r=216
7r=72
r=72%2F7

ANSWER: 72/7 = 10 2/7 = D

Answer by ikleyn(52794) About Me  (Show Source):
You can put this solution on YOUR website!
.
The circle with center O is inscribed in kite ABCD. Find the length of the radius of the circle, in cm.
https://ibb.co/94Z8wrZ
~~~~~~~~~~~~~~~~~~


            I will show you another solution. 


The area of the right-angled triangle ABC is half the product of its legs  %281%2F2%29%2A24%2A18 cm^2.

The area of the right-angled triangle ADC is the same %281%2F2%29%2A24%2A18 cm^2.


So, the area of the kit is simply  24*18  cm^2.



    For any convex polygon circumscribed around a circle, 
    its area is half the product the radius of the circle 
    by the perimeter of the polygon

        area of the polygon = %281%2F2%29%2Ar%2AP.



In our case, the perimeter of the kite is 2*(24+18) = 2*42 = 84 cm.


So we have

    area of the kite = 24*18 = %281%2F2%29%2Ar%2A84 = 42r,


which gives  r = %2824%2A18%29%2F42 = %284%2A18%29%2F7 = 72%2F7 = 10 2%2F7 cm.    ANSWER

Solved.