Question

In the below figure, □ABCD is a parallelogram. It circumscribes the circle with centre T. Point E, F, G, H are touching points. If AE=4.5, EB=5.5, find AD.

Answer 1

Hint: Given ABCD is a parallelogram. Opposite sides of a parallelogram are equal in length. Tangents drawn from an external point to a circle are congruent. Using these two statements find the length of AD.

Complete step-by-step answer:

We are given that ABCD is a parallelogram with a circle circumscribed in it. T is the centre of the circle and the lengths of AE, EB is 4.5, 5.5 respectively.

We have to find the length of AD.
We know that, as ABCD is a parallelogram the opposite sides are equal in it which means AB=CD and AD=BC
When tangents are drawn from an external point to a circle, then the tangents are congruent which means their lengths are equal and the angles they make with the circumference of the circle are equal.
The tangent drawn from external point A to the circle meets the circle at E and H.
This means $ AE = AH \to eq(1) $
The tangent drawn from external point B to the circle meets the circle at E and F.
This means $ BE = BF \to eq(2) $
The tangent drawn from external point C to the circle meets the circle at G and F.
This means $ CG = CF \to eq(3) $
The tangent drawn from external point D to the circle meets the circle at G and H.
This means $ DG = DH \to eq(4) $
Opposite sides are equal which means AB=CD and AD=BC
Add the equations 1, 2, 3, 4 we get
 $
  AE + BE + CG + DG = AH + BF + CF + DH \\
  AE + BE = AB \\
  CG + DG = CD \\
  AH + DH = AD \\
  BF + CF = BC \\
   \to AB + CD = AD + BC \to eq(5) \\
  $
AB=CD, This results $ AB + CD = AB + AB = 2AB $
AD=BC, This results $ AD + BC = AD + AD = 2AD $
Substitute the above values in equation 5.
 $
  2AB = 2AD \\
  AB = AD \\
  AB = AE + EB \\
  AD = AE + EB \\
  AE = 4.5 \\
  EB = 5.5 \\
  AD = 4.5 + 5.5 \\
  AD = 10 \\
  $
Therefore, the length of AD is 10 units.

Note: Circumscribing means to enclose within the bounds, just touching the corner points but never crossing them. A rectangle is also a parallelogram if its every angle is 90 degrees.