Answer

Verified

427.8k+ views

**Hint:**We use the property of tangents that tangents from an external point to the circle are equal in length. Calculate the lengths from the given measurements and use the concept of radius making the right angle at the point of contact with the tangent to make OPDS a square.

* The radius of the circle is perpendicular to the tangent at the point of contact.

* Tangents drawn from an external point to a circle are equal in length.

**Complete step-by-step solution:**

We have a quadrilateral ABCD where AD is perpendicular to DC, so \[\angle D = {90^ \circ }\]

We are given the lengths \[BC = 38\]cm,\[BQ = 27\]cm and \[DC = 25\]cm

Since BQ and BR are tangents from point B to the circle

\[ \Rightarrow BQ = BR\]

\[ \Rightarrow BR = 27\]cm

Also, \[RC = BC - BR\]

\[ \Rightarrow RC = 38 - 27\]cm

\[ \Rightarrow RC = 11\]cm

Since, RC and SC are tangents from point C to the circle

\[ \Rightarrow SC = RC\]

\[ \Rightarrow SC = 11\]cm (as \[RC = 11\]cm)

Now we know \[DC = 25\]cm

Then \[DS = DC - SC\]

\[ \Rightarrow DS = 25 - 11\]cm

\[ \Rightarrow DS = 14\]cm

We know the radius of the circle is perpendicular to the tangent at the point of contact.

So, \[OP \bot AD,OS \bot DC\]

Since all angles of quadrilateral OPDS are right angles, then it can be a rectangle or a square.

But since a rectangle has opposite sides equal in length, OPDS cannot be a rectangle as adjacent sides OP is equal to OS and PD is equal to DS.

So, OPDS must be a square.

Now we know a square has all sides of equal length, then \[OP = PD\]

\[ \Rightarrow OP = 14\]cm

So, the radius of the circle is 14cm

**\[\therefore \]Option B is the correct answer.**

**Note:**Students many times get confused between the pair of tangents that are equal and end up doing wrong calculations. Keep in mind we take the vertices of the triangle as the external points and the tangents are from that point to the circle, always write first the point associated with the equal tangents and then proceed with the solution. Also, the point of contact means the point where the tangent touches the circle.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Which are the Top 10 Largest Countries of the World?

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

How do you graph the function fx 4x class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Change the following sentences into negative and interrogative class 10 english CBSE