Question

A village Panchayat constructed a circular tank to serve as a bird bath. A fencing as made in the shape of a quadrilateral . Sides of the quadrilateral AB, BC,CD and DA touched the circle at E, F, G, H respectively. Find AD if AB=5 m , CD=6m , and BC=7m.

Answer 1

length of AD = 4 cm

Given:

• AB = 5 cm,
• BC = 7 cm and
• CD = 6 cm

To find: Length of AD

As shown in the figure, AB, BC, CD and AD are the tangents to the circle which touches the circle at points P, Q, R and S respectively.

Therefore,

• AP = AS,
• DS = DR,
• CR = QC,
• BQ = BP

Let

• AP = x cm so,
• BP = 5−x cm

thus,

• AP = AS = x cm and
• BP = BQ = (5−x) cm

Now BC = 7 cm (Given)

So,

•     BQ + QC = 7 cm  

• ⇒  QC + (5 − x) = 7

• ⇒  QC = 7 − 5 + x = (2 + x) cm

Thus,

•    CR = (2 +x) cm

Now,

•    CD = 6 cm (Given)

• ⇒ DR + CR = 6 cm

• ⇒ DR + (2 + x)= 6

• ⇒ DR = 6−2−xDR = (4−x) cm

Thus, SD = DR = (4−x) cm

Now, AD = AS + SD = x + 4−x = 4 cm

Hence, length of AD = 4 cm