Let ABC be a triangle in which AB AC Let L be the locus of points X inside or on the triangle such t...........

Quantitative Aptitude (CDS) Coordinate Geometry

Let ABC be a triangle in which AB =AC. Let L be the locus of points X inside or on the triangle such that BX =CX. Which of the following statements are

correct?

1. L Is a straight line passing through A and in-centre of triangle ABC is on L.

2. L is a straight line passing through A and orthocentre of triangle ABC is a point on L.

3. L is a straight line passing through A and centroid of triangle ABC is a point on L.

Select the correct answer using the code given below:

1 and 2 only

2 and 3 only

1 and 3 only

1, 2 and 3

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Explanation:

Locus of the point X is L.
Here L is line segment AD.
Now, Δ AXB ≅ Δ AXC (By SSS)
∴ ∠BAX =∠CAX (Corr. Angles)
⇒ AX and hence AD is the bisector of ∠BAC.
Hence in centre of the ΔABC lies on AD i.e., L                         (statement 1 is correct)
Since AB = AC, Find AD is the bisector of ΔABC
Now ΔXBD ≅ ΔXCD (By SSS)
∠XDB = ∠XDC = 90°
Hence AD is the perpendicular bisector of BC.
Therefore, orthocentre of the ΔABC lies on AD i.e, L.                 (Statement 2 is correct)
Since D is the mid-point of BC, therefore AD is the medium.
Hence centroid of the ΔABC lies on AD i.e., L.                           (Statement 3 is correct)