NCERT Solutions for Class 7th Maths Chapter 6 Construction of a 90° Angle at a Given Point — In-text Questions
Book page 141 Updated on2026-09-19
Q1.
Can we extend the method of constructing the perpendicular bisector to construct a 90° angle at any point on a line? Draw a line and mark a point O on it. Construct a 90° angle at point O.
Answer
Yes. Turn the problem around: instead of being given a segment and finding its midpoint, start from the midpoint and manufacture a segment.
Draw the line and mark O on it. Extend the line on both sides of O.
Open the compass to any radius. With centre O, cut the line at X on one side and Y on the other. Now OX = OY, so O is the midpoint of XY.
With a larger radius, draw arcs from X and from Y that cross above the line at A.
Join A to O. Then ∠AOX = ∠AOY = 90°.
O is made the midpoint of XY first. The perpendicular bisector of XY then has to pass through O.
Why it happens: AX = AY makes A a point of the perpendicular bisector of XY, and OX = OY makes O another point of it. So the line AO is that perpendicular bisector — and a perpendicular bisector meets the segment at a right angle. The 90° lands at O because we chose O to be the midpoint.
Q2.
Find a segment of this line for which O is the midpoint.
Answer
Extend the line on both sides of O. Set the compass to any convenient opening, put the point at O, and cut the line on both sides — at X and at Y.
Same compass opening on both sides ⇒ OX = OY
X, O, Y are on one straight line
⇒ O lies between X and Y and splits XY into two equal parts
⇒ O is the midpoint of XY
Why it happens: we cannot bisect a point, but we can build a segment around it. Any opening of the compass will do — a big opening gives a long XY, which makes the arcs in the next step cross more sharply and the drawing more accurate.
Tip: XY does not have to be any particular length. Only OX = OY matters.
Q3.
In this case, do we need to draw two pairs of intersecting arcs to get the perpendicular bisector of XY?
Answer
No — one pair is enough.
Point 1 on the perpendicular bisector: O (because OX = OY)
Point 2 on the perpendicular bisector: A from the single pair of arcs
Two known points fix the whole line
⇒ join A to O and the line is drawn
Why it happens: in the original construction we had to hunt for two points, so we drew two pairs of arcs. Here one of the two points is handed to us — O is on the bisector the moment we make it the midpoint. So only one more point is needed, and one pair of arcs supplies it.
Tip: drawing the second pair below the line does no harm, and it gives a longer line to rule along. It is extra accuracy, not extra logic.