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Practical Geometry — MATHS STD 6 — Question

Gujarat BoardEnglish MediumSTD 6MATHSPractical Geometry5 Marks
Question
Draw an angle and label it as $\angle\text{BAC}.$ Draw its bisector ray $AX$ and take a point $P$ on it. From $P$ draw line segments $PM$ and $PN,$ such that $\text{PM}\perp\text{AB}$ and $\text{PN}\perp\text{AC},$ where $M$ and $N$ are respectively points on rays $AB$ and $AC.$ Measure $PM$ and $PN.$ Are the two lengths equal$?$
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Answer

$i.\ $Draw $\angle\text{BAC}$ on the line segment $AC.$
With a convenient radius and $A$ as centre, draw an arc from $AB$ and $AC$.
$ii.\ $The points where arc cuts $AB$ and $AC,$ take both points as centres and draw two small arcs intersecting at $X.$ Now, draw $AX.$
$iii.\ $Take a point $P$ on the ray $AX.$
$iv.\ $Take a convenient radius with $P$ as centre and construct an arc intersecting the line segments $AB$ at $T$ and $Q$ and $AC$ at $R$ and $S,$ respectively.
$v.\ $Using the same radius and with $T$ and $Q$ as centres, construct two arcs intersecting at $G$ on the other side.
$vi.\ $Using the same radius and with $R$ and $S$ as centres, construct two arcs intersecting at $H$ on the other side.
$vii.\ $Join $PG$ and $PH,$ which intersects $AB$ and $AC$ at $M$ and $N,$ respectively.
On measuring $PM$ and $PN$ using a ruler, we find that both are equal.

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