Draw the perpendicular bisector of $ \overline{X Y} $ whose length is $ 10.3 \mathrm{~cm} $. (a) Take any point $ \mathrm{P} $ on the bisector drawn. Examine whether $ \mathrm{PX}=\mathrm{PY} $. (b) If $ \mathrm{M} $ is the mid point of $ \overline{\mathrm{XY}} $, what can you say about the lengths $ \mathrm{MX} $ and $ \mathrm{XY} $ ?
We have to draw the perpendicular bisector of $\overline{XY}$ whose length is $ 10.3\ cm$ and answer the given questions.
Solution:
Steps of construction:
(i) Let us draw a line segment $\overline{XY}$ of length $ 10.3\ cm$ .
(ii) Now, with compasses take a measure greater than half of the length $\overline{XY}$
(iii) Now, by pointing the pointer of the compasses at point X and point Y respectively. let's draw two arcs above the line segment $\overline{XY}$ cutting each other and point it as C.
(iv) Similarly, by pointing the pointer of the compasses at point X and point Y respectively. let's draw another two arcs below the line segment $\overline{XY}$ cutting each other and point it as D.
(iv) Then, let's draw a line segment joining point C and point D.
(v) Therefore, the perpendicular bisector of the line segment $\overline{XY}$ is formed.
(a) Let us take any point P on the bisector CD.
Since $\overline{CD}$ is the perpendicular bisector of $\overline{XY}$ any point lying on $\overline{CD}$ will be at the same distance from point X and point Y of line segment $\overline{XY}$.
Therefore,
We may measure the length of $PX = PY$.
(b) As M is the midpoint of $\overline{XY}$. The perpendicular bisector $\overline{CD}$ passes through point M.
Therefore,
The length of $\overline{XY}$ is twice $\overline{MX}$.
That is $\overline{MX}$ is $\frac{1}{2}\overline{XY}$.
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