Xavier Aptitude Test 2020 Paper

Question : 64 of 100

Marks: +1, -0

$\XYZ$ is an equilateral triangle, inscribed in a circle. P is a point on the arc $\YZ$ such that $\X$ and $\P$ are on opposite sides of the chord $\YZ$ . Which of the following MUST always be true?

$\XZ + \YP = \XY + \PZ$
$\XP = \XY$
$\XP + \PZ = \XY + \YP$
$\XP = \YP + \PZ$
$\XP = \XY + \YZ$

Validate

Solution:

If we join

\YP

and

\PZ, \XYPZ

will become a cyclic quadrilateral.

\YZ

and

\XP

will be diagonals of this quadrilateral.
Ptolemy's theorem states that the product of the length of the diagonals is equal to the sum of products of measures of the pairs of opposite sides.
As per this theorem for the quadrilateral :

\XY × \PZ + \XZ × \PY = \YZ × \XP

For the equilateral triangle.
Now, in equilateral triangle

\XYZ, \YZ = \XZ = \XY

.
Hence equation 1 becomes,

\XP = \YP + \PZ

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