Draw an angle and label it as ∠BAC. Draw its bisector ray AX and take a point P on it. From P draw line segments PM and PN, such that PM ⊥ AB and PN ⊥ 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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Solution

(i) Draw ∠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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