ABC is an equilateral triangle and X Y and Z are the points on BC CA and AB respectively such that B...........

Quantitative Aptitude (CDS) Coordinate Geometry

ABC is an equilateral triangle and X, Y and Z are the points on BC, CA and AB respectively, such that BX = CY = AZ. Which of the following is/are correct?

1. XYZ is an equilateral triangle.

2. $\Delta$ XYZ is similar to DABC.

Select the correct answer using the code given below.

Only 1

Only 2

Both 1 and 2

Neither 1 nor 2

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Explanation:

In an equilateral $\Delta$ ABC,
AB = BC = CA and $\angle$ B = $\angle$ C = $\angle$ A
Given that, BX = CY = AZ
Now, in $\Delta$ XYC, $\Delta$ ZYA and $\Delta$ XZB
BX = CY = AZ …(i)
=> (BC − XC) = (AC − AY) = AB − BZ
=> XC = AY = BZ …(ii)
and $\angle$ B = $\angle$ C = $\angle$ A …(iii)
From Eqs. (i), (ii) and (iii), we get $\Delta$ XYC, $\Delta$ ZYA and $\Delta$ XZBare congruent triangle.
$\therefore$ XY = YZ = XZ
So, $\Delta$ XYZis an equilateral triangle and DXYZis similar to $\Delta$ ABC.
So, both statements are correct.