Question

Draw a line segment AB and by ruler and compasses, obtain a line segment of length \[\dfrac{3}{4}(AB)\].

Answer 1

Hint: For intersecting or we can say that for constructing any diagram we must mark the compass point and ruler points at the exact points because even a mm difference in marking the points will change our intersecting point.

Complete step by step solution:
Now let us solve this question step by step so that you can easily get the solution.
Step 1. First of all, draw a straight - line segment let say AB with the help of a ruler.

Step 2. Now draw an arc by taking the point A as centre and opening the compass more than half of AB on both above and below the line AB.

Step 3. Similarly draw the arc again with the same length of compass opening but this time we would take B as centre.
Step 4. Name these arc points as M and N and now join M to N.

Step 5. Now MN must intersect AB at some point so name that point as O.

Step 6. Now again draw an arc on both sides of AO by taking A as centre and radius more than half of AB.
Step 7. By taking O as a centre with the same radius, draw another arc which cuts the arc drawn in step 6.
Step 8. Name these arc points as P and Q and now join P to Q .

Step 9. This PQ must also intersect AB at some point so name that point as X.
Step 10. Bisect MN again and mark the point of bisection as X.
So, now we have
AX = \[\dfrac{1}{4}(AB)\]
OX = \[\dfrac{1}{4}(AB)\]and OB = \[\dfrac{1}{2}(AB)\]
Therefore XB = \[\dfrac{1}{4}(AB)\; + \;\dfrac{1}{2}(AB)\; = \;\dfrac{3}{4}(AB)\]
Thus, XB is the required line segment of length \[\dfrac{3}{4}(AB)\].

Note: whenever we come up with this type of question we must bisect the given line segment in ½ portion at first of all then we can bisect one of the bisected portions ( here it is OA or OB ) . and after bisecting the half potion we will get the desired length that we want.