Question

Question 86

Bisect a right angle, using ruler and compass. Measure each part. Bisect each of these parts. What will be the measure of each of these parts?

Answer 1

Steps of construction are as follows:

Step I: Construct an angle, $\angle ABC={90}^0$

Step II: With B as centre, using a compass, draw an arc which cuts both rays of $\angle B$ at P and Q

Step III: With P as centre, draw (in the interior of $\angle B$ an arc, whose radius is more than half of PQ.

Step IV: With Q as centre and the same radius, draw another arc in the interior of $\angle B$. Let the two arcs intersect at D. Join $\overline{BD}$ cutting arc PQ at L. Then, $\overline{BD}$ divides the $\angle ABC$ into two equal parts.

Step V: Now, taking P and L as centre having radius more than half of PL, draw two arcs respectively, which cut each other at R.

Step VI: Join $\overline{BR}$, which divides $\angle CBD$ into two equal parts.

Step VII: Now, taking Q and L as centre, having radius more than half the length of QL, draw two arcs respectively, which cut each other at M.

Step VIII: Join $\overline{BM}$ which divide $\angle ABD$ into two equal parts.

Thus, $\overline{BM}$,$\overline{BD}$and $\overline{BR}$ divide $\angle ABC$ into four equal parts, i.e.

$\angle ABM=\angle MBD=\angle DBR=\angle RBC={22.5}^o$