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Circles

Question
CBSEENMA9002770
Wired Faculty App

In triangle ABC, points M and N on sides AB and AC respectively are taken so that 

AM equals 1 fourth AB space and space AN equals 1 fourth AC comma space Prove space that space MN equals 1 fourth

Solution
Given: In triangle ABC, points M and N on the sides AB and AC respectively are taken so that
AM space equals 1 fourth AB space and space AN equals 1 fourth A C
To prove: MN equals 1 fourth BC.

Construction: Join EF where E and F are the middle points of AB and AC respectively.
Proof: Y E is the mid-point of AB and F is the mid-point of AC.

therefore space space EF space parallel to space BC space and space space space space space space EF equals 1 half BC space space space space space space space space space space space space space... left parenthesis 1 right parenthesis space space space Now comma space space space space space space space space space space space space space space space space space space space AE equals 1 half AB space space space and space space space space space space space space space space space space space space space space space space space space AM equals 1 fourth space AB therefore space space space space space space space space space space space space space space space space space space space space space space space space space AM space equals space 1 half AE Similarly comma space space space space space space space space space space space space space space space space AN space equals space 1 half space AF
rightwards double arrow space space M and N are the mid-points of AE and AF respectively.
therefore space space MN space parallel to space EF space and space MN space equals space 1 half space EF equals 1 half open parentheses 1 half BC close parentheses
                                                             | From (1)
                                             equals space 1 fourth space BC.
                           
 

Some More Questions From Circles Chapter

If the diagonals of a parallelogram are equal, then show that it is a rectangle.

Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

 Show that the diagonals of a square are equal and bisect each other at right angles.

Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.

Diagonal AC of a parallelogram ABCD bisects ∠A (see figure). Show that:
(i)    it bisects ∠C also
(ii)    ABCD is a rhombus.

ABCD is a rhombus. Show that diagonal AC bisects ∠A as well as ∠C and diagonal BD bisects ∠B as well as ∠D.

ABCD is a rectangle in which diagonal AC bisects ∠A as well as ∠C. Show that (i) ABCD is a square (ii) diagonal BD bisects ∠B as well as ∠D.

In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see figure). Show that:

(i)    ∆APD ≅ ∆CQB
(ii)   AP = CQ
(iii)  ∆AQB ≅ ∆CPD
(iv)  AQ = CP
(v)   APCQ is a parallelogram.

ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD respectively (see figure). Show that:
(i) ∆APB ≅ ∆CQD
(ii) AP = CQ.

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