Circle in a Quadrilateral

dtweah wrote:A circle is inscribed in quadrilateral ABCD where the circle is tangent to each side of the quadrilateral , with AB = 16 and CD = 10. What is the perimeter of the quadrilateral?

(a) 50
(b) 52
(c) 54
(d) 56
(e) 58

If the circle touches the sides AB, BC, CD, and DA in P, Q, R, and S, respectively, then let DR = DS = x such that CR = CQ = 10 - x, and let BP = BQ = y such that AP = AS = 16 - y.

Now, AD + BC = (AS + DS) + (BQ + CQ) = (16 - y + x) + (y + 10 - x) = 26

Hence, the perimeter = AB + CD + AD + BC = 16 + 10 + 26 = [spoiler]52[/spoiler].

[spoiler]B[/spoiler]

Sanju

it seems a kind of theorem...

perimater is twice of sum of two opposite sides... kind of isn'tit...

are these sides(opposite) parallel too... ?

is there any SAYING or any rule regarding cyclic quadilateral..?

girish3131 wrote:Sanju

it seems a kind of theorem...

perimater is twice of sum of two opposite sides... kind of isn'tit... NO! Not the least!!

are these sides(opposite) parallel too... ? Not necessary

is there any SAYING or any rule regarding cyclic quadilateral..?

From a point outside of a circle, exactly two tangents can be drawn to the circle, and the two tangents measure same.

A quadrilateral whose all four vertices are on a circle is termed as a cyclic quadrilateral. The opposite angles in a cyclic quadrilateral are supplementary.

girish3131 wrote:Sanju

it seems a kind of theorem...

perimater is twice of sum of two opposite sides... kind of isn'tit...

are these sides(opposite) parallel too... ?

is there any SAYING or any rule regarding cyclic quadilateral..?

sorry girish3131,

I think you were correct at it, but since you didn't mention the thing completely, I had to deny it. You should have mentioned a cyclic quadrilateral, at least.

For others...

The perimeter of a cyclic quadrilateral is twice the sum of its one pair of opposite sides.