Home
Class 10
MATHS
Let M be a point of contact of two inter...

Let M be a point of contact of two internally touching circles. Let line AMB be their common tangent. The chord CD of the bigger circle touches the smaller circle at the point N. Chord CM and chord DM of the bigger circler intersect the smaller circle at the points P and R respectively.
Prove that `/_ CMN ~= /_ DMN`.
Construction `:` Draw seg NR.

Text Solution

Verified by Experts

CD si a tangent to the smaller circle at the point N and NR is a chord of the smaller circle.
`:. /_ DNR ~= /_ NMR ` …(Tangent secant theorem ) …(1)
AB is a tangent to the lager circle at the point M and CM is th echord of the larger circle.
`:. /_ CMA ~= /_ CDM ` ...(Tangent secant theorem )
i.e., `/_ CMA ~= /_D ` ...(2)
AB is a tangent to the smaller circle at the point M and NM is a chord of the smaller circle .
`:. /_ NMA ~= /_ NRM ` ....(Tangent secant theorem ) ...(3)
For `Delta NDR, /_ NRM ` is the exterior theorem,
`/_ NRM = /_ D + /_ DNR ` ...(4)
`:. /_ NMA = /_ CMA + /_ NMR` ........[From (3), (2) and (1) ] ....(5)
`/_ NMA = /_ NMC + /_ CMA ` ....(Angle addition postulate ) ...(6)
From (5) and (6)
`/_ NMC = /_ NMR `
`:. /_NMC = /_ NMD ` ...( `/_ NMR ` is the same as `/_ NMD `)
i.e., `/_ CMN ~= /_ DMN `.
Promotional Banner

Topper's Solved these Questions

  • CIRCLE

    NAVNEET PUBLICATION - MAHARASHTRA BOARD|Exercise ASSIGNMENT 4.1|10 Videos

  • CIRCLE

    NAVNEET PUBLICATION - MAHARASHTRA BOARD|Exercise ASSIGNMENT 4.2|6 Videos

  • CIRCLE

    NAVNEET PUBLICATION - MAHARASHTRA BOARD|Exercise 4.4 (1 mark each)|9 Videos

  • CHALLENGING QUESTIONS

    NAVNEET PUBLICATION - MAHARASHTRA BOARD|Exercise SECTION 3 (MODEL QUESTION PAPER FOR PRACTICE ) Solve any one of the following subquestions :|1 Videos

  • COORDINATE GEOMETRY

    NAVNEET PUBLICATION - MAHARASHTRA BOARD|Exercise Assignment 6.5|13 Videos

Similar Questions

Explore conceptually related problems

In the figure, M is the point of contact of two internally touching circles. The chord CD of the bigger circle touches the smaller circle at point N. Line AMB is their common tangent. Prove MN bisects /_CMD

In the figure, two circles touch internally at point P. chord AB of the larger circle intersects the smaller circle in C and D. Prove /_CPA cong /_DPB .

In the given figure, centre of two circles is O. Chord AB of bigger circle intersects the smaller circle in points P and Q . Show that AP = BQ.

Two concentric circles are of diameter 30cm and 18cm. Find the length of the chord of the larger circle which touches the smaller circle.

Two concentric circles are of radii 5cm and 3cm. Find the length of the chord of the larger circle which touches the smaller circle.

Two concentric circles are of radii 5cm and 3cm. Find the length of the chord of the larger circle which touches the smaller circle.

Two concentric circles are of diameters 30cm and 18cm. Find the length of the chord of the larger circle which touches the smaller circle.

In two concentric circles, a chord of length 8cm of the larger circle touches the smaller circle. If the radius of the larger circle is 5cm then find the radius of the smaller circle.