Question

Using ruler and compasses only, construct a $\Delta ABC$ having $AB=47\text{cm}$, $AC=34\text{cm}$ and $\angle BAC={75}^{\circ }$. Draw the perpendicular bisector of $BC$ and the bisector of $\angle BAC$. If the perpendicular bisector and the angle bisector meet at a point $M$, measure $\angle BMC.$

Answer 1

1. Draw a line segment measuring 47 cm and name it as AB.

2. To Construct 75 deg:

Start with two angles, ∠RPQ and ∠DAB.

The goal is to add ∠RPQ to ∠DAB.

Set the compasses on A, and adjust to any convenient width.

Make an arc across both legs of the angle, creating points J and K.

Set the compasses on A and make an arc to the left of, and across BC.

Set the compasses on K and adjust the width to point J.

Set the compasses on L and make an arc across the previous one, creating point E.

Draw a line from A, through E. Measure 34cm on the compass and cut on the line segment AE

The angle CAB which now measures 75 deg. Join B and C and triangle CAB is formed.

3. To draw a perpendicular bisector:

Measure more than half of BC on the compass and cut arcs on either side of BC keeping the compass at B.

Similarly, with the same radius, keeping the compass at C cut the arc drawn in the previous step.

The line joining the intersection of these arcs is the perpendicular bisector of BC.

4. Construct the bisector of 75 deg.

5. The point where the angular bisector and perpendicular bisector meet is the point M.

6. BMC is 106 deg.