Question

Draw a line segment $\text{PQ = 8}\text{.4 cm}$ and divide into four equal parts using ruler and compass.

Answer 1

Hint: We will draw a segment of approximately the same length as PQ from the point P. Then we will use the compass to make 4 arcs on this new segment is a certain manner. We will draw another segment parallel to the new segment using the compass. We will make 3 arcs on this segment. Then we will join the corresponding points on the two parallel segments to make 4 equal parts of segment PQ.

Complete step-by-step solution:
We have the given segment PQ with a length of 8.4 cm. Let us draw a segment from point P making some angle with segment PQ. We will label this segment as PR. With the compass, we will make 4 equal length arcs on segment PR. It should look roughly like the figure below.

Next, we will set the compass's length to be the length between point A and point Q. We will draw an arc of this length from point P just below it. Now we will set the length of the compass to be the length of segment PA and draw an arc from point Q so that it intersects the arc drawn just below point P. Let us call the intersection of these two arcs to be point B. Let us join points Q and B to form the segment QB.
Now, we will set the length of the compass to be the same while we were drawing four arcs on segment PR. Using this length on the compass, we will make three arcs on the segment QB starting from point Q. It should look similar to the following figure,

Now, we will join the corresponding points on segment PA and segment QB in the following manner,

We can see that at $\text{PU=UV=VW=WQ}$. Hence, we have made four equal parts of the segment PQ using the ruler and the compass,


Note: The figures above are not actually to scale. They are just to give an idea of what to expect while following the procedure of this geometric construction. It is necessary that we make sure to use the compass properly. Otherwise, a slight change in the measurement will lead to incorrect construction.