Use app×
Join Bloom Tuition
One on One Online Tuition
JEE MAIN 2025 Foundation Course
NEET 2025 Foundation Course
CLASS 12 FOUNDATION COURSE
CLASS 10 FOUNDATION COURSE
CLASS 9 FOUNDATION COURSE
CLASS 8 FOUNDATION COURSE
0 votes
4.3k views
in Circle by (46.9k points)
closed by

In the adjoining figure, the circles with centres A and B touch each other at E. Line l is a common tangent which touches the circles at C and D respectively. Find the length of seg CD if the radii of the circles are 4 cm, 6 cm.

1 Answer

+1 vote
by (47.6k points)
selected by
 
Best answer

Construction : Draw seg AF ⊥ seg BD.i. The circles with centres A and B touch each other at E. [Given] 

∴ By theorem of touching circles, 

A – E – B

∴ ∠ACD = ∠BDC = 90° [Tangent theorem] 

∠AFD = 90° [Construction] 

∴ ∠CAF = 90° [Remaining angle of AFDC] 

∴ AFDC is a rectangle. [Each angle is of measure 900] 

∴ AC = DF = 4 cm [Opposite sides of a rectangle] 

Now, BD = BF + DF [B – F – C]

∴ 6 = BF + 4 BF = 2 cm 

Also, AB = AE + EB 

= 4 + 6 = 10 cm 

[The distance between the centres of circles touching externally is equal to the sum of their radii]

ii. Now, in ∆AFB, ∠AFB = 90° [Construction] 

∴ AB2 = AF2 + BF2 [Pythagoras theorem] 

∴ 102 = AF2 + 22 

∴ 100 = AF2 + 4 

∴ AF2 = 96 

∴ AF = √96 [Taking square root of both sides] 

= √(16 x 6)

= 4√6 cm 

But, CD = AF [Opposite sides of a rectangle] 

∴ CD = 4√6 cm

Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students.

...