Question

The measurement of sides AB,BC,CD & DA of a trapezium ABCD are 20cm,10cm,36cm & 10cm respectively. If AB||CD, then show that it is a cyclic trapezium​

Answer 1

Step-by-step explanation:

Draw two altitudes from A and B onto CD and call the intersection P and Q.

⇒ Then Two rigjht triangles will be formed, △APD and △BQC.

⇒ △BQC has sides x,h,20.

⇒ and △APD has sides x−8,h,16.

⇒ Applying Pythagoras theorem in △APD and △BQC,

⇒ x

2

+h

2

=20

2

⇒ x

2

+h

2

=400 ------- (1)

⇒ (x−8)

2

+h

2

=16

2

⇒ x

2

−16x+64=256 -------- (2)

Now subtracting equation (1) and (2) we get,

⇒ 16x−64=400−256

⇒ 16x=144+64

⇒ 16x=208

⇒ x=13

This shows QC=13 and QD=5

So that the longer diagonal must be AC rather than BD.

Now to find the length of AC, look at the right triangle APC, with sides 23,h,d.

⇒ d

2

=h

2

+23

2

⇒ d

2

=20

2

−13

2

+23

2

⇒ d

2

=400−169+529

⇒ d

2

=760

⇒ d=

760

cm