Question

ABCD is a trapezium. Sides AB & CD are parellel to each other. Side AD is perpendicular to side AB. A circle with centre O touches all the sides internally.
Find the area and perimeter of the quadrilateral ABCD if OB = 6cm &
OC = 8cm.​

Answer 1

Given : ABCD  trapezium.  AB parallel   CD . AD  perpendicular to AB.  Circle with  centre O touches all the sides internally OB = 6cm &  OC = 8cm.​.

To find : area and perimeter of the quadrilateral ABCD

Solution:

referring attached diagram

y ² + r² = 6²

x² + r² = 8²

Adding both

x² + y² + 2r²  = 100    Eq 1

Subtracting

x² - y²  = 28            

=>(x + y)(x - y)  = 28     Eq 2

Let Draw a perpendicular from B at CD

then we get

(2r)²  + (x - y)²  = (x + y)²

=> 4r² + x² + y² -2xy = x² + y² + 2xy

=> 4r² = 4xy

=> r² = xy

putting this in Eq 1

=> x² + y² + 2xy  = 100

=> (x + y)² = 10²

=> x + y = 10

Putting this in eq 2

=> (10)(x - y) = 28

=> x  - y  = 2.8

x + y = 10

x - y = 2.8

=> 2x = 12.8

=> x = 6.4

& y  = 3.6

r² = xy

=> r² = 6.4 * 3.6

=> r =  4.8

Perimeter  =  4r  + 2x + 2y

= 4(4.8) + 2(6.4) + 2(3.6)

= 39.2 cm

Area =  (1/2) ( r + x + r + y) ( 2r)

=  (19.6)(4.8)

= 94.08 cm²

Perimeter of Trapezium = 39.2 cm

Area of Trapezium = 94.08 cm²

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