Question

A quadrilateral of perimeter 126 cm is circumscribed about a circle. Three of its sides taken in order are in the ratio 5 : 8 : 9. What is the length of the fourth side?

Answer 1

Answer:

27 cm is the length of the fourth side.

Step-by-step explanation:

A quadrilateral of perimeter 126 cm is circumscribed about a circle.

Three of its sides taken in order are in the ratio 5 : 8 : 9

Please see the attachment for figure.

Let three sides of quadrilateral be 5x, 8x and 9x

AB=5x  ,  BC=8x   and   CD=9x

The length of tangents to the circle from an external point are equal.

AP=AQ=a

QB=BR=5x-a

RC=SC=3x+a

DS=DP=6x-a

Perimeter of quadrilateral ABCD = 126

AP+AQ+QB+BR+RC+SC+DS+DP=126

2(AP+BR+SC+DP)=126

a+5x-a+3x+a+6x-a=63

14x=63

Length of fourth side = 126 - 22x

Length of fourth side = $126-22\cdot \frac{63}{14}$

Length of fourth side = 126 - 99 = 27 cm

Thus, 27 cm is the length of the fourth side.