   Question Answers

# In figure, CP and CQ are tangents to the circle with center O. ARB is another tangent touching the circle at R. If CP = 12 cm, and BC = 8 cm, then find the length of BR.   Hint: Read the question carefully, and consider all the given information as it leads to solution.
Tangent to a circle is a line that intersects the circle at only one point.
The property of Pair of tangents to a circle drawn from an external point is the key to solve this question.

Complete step by step solution:
Step 1: Draw the labeled diagram carefully: Figure 1: circle with center O
Step 2: Given that:
Line CP, CQ, and AB are tangents to the circle with center O.
Length CP = 12 cm, BC = 8 cm
Step 3: State the theorem related to Pair of tangents to a circle drawn from an external point:
Theorem 1: The lengths of tangents drawn from an external point to a circle are equal. Figure 2: tangents PQ and PR to the circle with center O
Hence, length PQ = length PR
Step 4: apply the theorem
Here, line CP and CQ are tangents from external point C to the circle with center O.
$\Rightarrow$CP = CQ
Hence, CQ = 12 cm
Step 5: since, QBC is a straight line:
Therefore, CQ = BQ + BC
12 = BQ + 8
BQ = 12 – 8
$\Rightarrow$BQ = 4 cm. …… (1)
Step 6: apply the theorem
Here, line BQ and BR are tangents from external point B to the circle with center O.
$\Rightarrow$BQ = BR
Hence, BR = 4 cm (from (1))

Therefore, the required length of the BR is 4 cm.

Note:
Another important theorem of tangents and circle:
Theorem: The tangent at any point of a circle is perpendicular to the radius through the point of contact. Figure 3: tangent XY to the circle with center O.
Hence, $OP \bot PQ$
There is only one tangent at a point of the circle.
View Notes
CBSE Class 10 Maths Chapter 8 - Introduction to Trigonometry Formula  What are the Successor and Predecessor?  Area of a Sector of a Circle Formula  What are the Functions of the Human Skeletal System?  IMO Maths Olympiad Sample Question Paper for Class 6 to 10  CBSE Class 8 Maths Chapter 15 - Introduction to Graphs Formulas  CBSE Class 6 to 12 Maths Formulas - Important Math Formulas  IMO Maths Olympiad Previous Year Question Paper for Class 5 to 10  Group Theory in Mathematics  To Determine the Mass of Two Different Objects Using a Beam Balance  Important Questions for CBSE Class 10 Maths Chapter 8 - Introduction to Trigonometry  Important Questions for CBSE Class 6 English A Pact with The Sun Chapter 9 - What Happened to The Reptiles  Important Questions for CBSE Class 10 Maths Chapter 12 - Areas Related to Circles  Important Questions for CBSE Class 6 English A Pact with The Sun Chapter 6 - The Monkey and the Crocodile  Important Questions for CBSE Class 6 English A Pact with The Sun Chapter 10 - A Strange Wrestling Match  CBSE Class 7 Science Fibre to Fabric Worksheets  CBSE Class 6 Science Fibre to Fabric Worksheets  Important Questions for CBSE Class 8 Maths Chapter 15 - Introduction to Graphs  Important Questions with Answers for CBSE Class 6 to 12 - All Subjects  Important Questions for CBSE Class 6 English A Pact with The Sun Chapter 8 - A Pact with the Sun  CBSE Class 10 Hindi A Question Paper with Solutions  CBSE Previous Year Question Papers Class 10 Maths with Solutions  CBSE Class 10 Hindi A Question Paper 2020  Hindi A Class 10 CBSE Question Paper 2009  Hindi A Class 10 CBSE Question Paper 2015  CBSE Class 10 English Language and Literature Question Paper with Solutions  CBSE Class 10 Maths Question Paper 2017  CBSE Class 10 Maths Question Paper 2020  Hindi A Class 10 CBSE Question Paper 2007  Hindi A Class 10 CBSE Question Paper 2013  RD Sharma Class 11 Maths Solutions Chapter 24 - The Circle  RD Sharma Class 11 Solutions Chapter 24 - The Circle (Ex 24.2) Exercise 24.2  RD Sharma Class 11 Solutions Chapter 24 - The Circle (Ex 24.1) Exercise 24.1  NCERT Solutions for Class 10 Maths Chapter 8 Introduction to Trigonometry in Hindi  NCERT Solutions for Class 10 Maths Chapter 12 Areas Related to Circles in Hindi  RD Sharma Class 11 Solutions Chapter 24 - The Circle (Ex 24.3) Exercise 24.3  NCERT Solutions for Class 10 Maths Chapter 8 Introduction to Trigonometry  RS Aggarwal Class 11 Solutions Chapter-21 Circle  NCERT Solutions for Class 10 Maths Chapter 12 Areas Related to Circles  RD Sharma Class 12 Maths Solutions Chapter 16 - Tangents and Normals  