Maharashtra Board Class 6 Maths part 2 Chapter 17 Geometrical Constructions PDF Download

Geometrical Constructions

Can You Tell?

(1) When constructing a building, what is the method used to make sure that a wall is exactly upright? What does the mason in the picture have in his hand? What do you think is his purpose for using it?

(2) Have you looked at lamp posts on the roadside? How do they stand?

The Perpendicular

In the figure here, line l and line n intersect at point M.

Measure every angle formed at the point M.

If an angle between line l and line n is a right angle, we say that the lines are perpendicular to each other. This is written as 'line l ⊥ line n' in symbols. It is read as 'Line l is perpendicular to line n'.

Drawing A Perpendicular To A Line At A Point On The Line

(1) Using A Set Square

Draw line PQ. Take point R anywhere on the line.

Place the set square on the line in such a way that the vertex of its right angle is at point R and one arm of the right angle falls on line PQ.

Draw a line RS along the other arm of the right angle of the set square.

The line RS is perpendicular to the line PQ at R.

Teacher's Note

When we build a house, we use a set square to make sure the walls are straight up and down. This is just like how carpenters in India use this tool to make sure doors and windows are perfectly upright.

Exam Trick

Remember: Perpendicular means 90 degrees, like the corner of your book. When two lines meet at 90 degrees, they are perpendicular.

Points To Remember

Perpendicular lines meet at a right angle of 90 degrees.
The symbol ⊥ means perpendicular.
A set square has a right angle that helps us draw perpendicular lines.
We place the right angle of the set square at the point where we want the perpendicular line.

(2) Using A Protractor

Draw line RS. Take point M anywhere on the line.

In order to draw a perpendicular through M, place the centre of the protractor on point M, as shown.

Mark a point N at the 90° mark on the protractor.

Draw a line passing through points M and N.

The line MN is perpendicular to line RS at M.

Line MN ⊥ line RS.

Teacher's Note

A protractor is a tool that measures angles. We can use it to draw a 90-degree angle which is a perpendicular line. Just like using a measuring tape, we measure the angle and then draw the line.

Exam Trick

Remember: Find the 90° mark on the protractor and put a dot there. Then draw a line from the center point through that dot. That's your perpendicular line!

Points To Remember

A protractor has markings from 0° to 180°.
The centre of the protractor must be placed exactly on the point.
We mark the 90° position and draw a line through it.
This line will be perpendicular to the original line.

(3) Using A Compass

Draw line MN. Take point K anywhere on the line.

Place the compass point on point K. Draw two arcs on either side of point K to cut the line MN at equal distances from K. Name the points of intersection A and B respectively.

Place the compass point at A and, taking a convenient distance greater than half the length of AB, draw an arc on one side of the line.

Place the compass point at B and using the same distance, draw another arc to intersect the first one at T.

Draw a line passing through points K and T.

The line KT is perpendicular to line MN at K.

Line KT ⊥ line MN.

Think About It

Why must we take a distance greater than half of the length of AB? What will happen if we take a smaller distance?

Teacher's Note

Using a compass is like drawing circles. If we draw two circles that are big enough, they will cross each other. The crossing point helps us find the perpendicular line. This is how builders in India sometimes check if walls are straight.

Exam Trick

Remember: The compass distance must be more than half of AB, or the two arcs will not meet. If they don't meet, you cannot draw the perpendicular line.

Points To Remember

We draw equal arcs from points A and B on the line.
The compass distance must be the same for both arcs.
The compass distance must be more than half of AB.
The arcs meet at point T above or below the line.
The line through K and T is perpendicular to the original line.

Practice Set 39

1. Draw line l. Take any point P on the line. Using a set square, draw a line perpendicular to line l at the point P.

2. Draw a line AB. Using a compass, draw a line perpendicular to AB at the point B.

3. Draw line CD. Take any point M on the line. Using a protractor, draw a line perpendicular to line CD at the point M.

Drawing A Perpendicular To A Line From A Point Outside The Line

(1) By Folding The Paper

Draw a line MN on a paper.

Take a point P anywhere outside the line.

Keeping the line MN in view, fold the paper along the line MN.

Now fold the paper through point P in such a way that the part of line MN on one side of the fold falls on the part of line MN on the other side of the fold.

Unfold the paper. Name the point of intersection of the two folds Q. Draw the line PQ. This line falls on a fold in the paper.

Using a protractor, measure every angle formed at the point Q.

Line PQ is perpendicular to line MN.

Line PQ ⊥ line MN.

Teacher's Note

Folding paper is a fun way to understand perpendicular lines. When you fold the paper so that the line matches itself, the fold line becomes perpendicular. It's like folding a piece of cloth in half to find the middle line.

Exam Trick

Remember: When you fold the paper so that one part of the line matches the other part, the fold line is perpendicular to that line. This is why origami uses perpendicular lines to make beautiful shapes.

Points To Remember

Folding the paper along a line creates a perpendicular at the fold point.
The perpendicular line is formed where two folds meet.
We can use a protractor to check if the angle is 90 degrees.
This method works because folding matches both sides perfectly.

(2) Using A Set Square

Draw line XY. Take point P anywhere outside XY.

Place one of the arms of the right angle of a set square along the line XY.

Slide the set square along the line in such a way that the other arm of its right angle touches point P. Draw a line along this side, passing through point P. Name the line PS.

Measure the angles to verify that the line is a perpendicular.

Teacher's Note

This method is very useful in real life. When carpenters want to make a perpendicular from a point to a wall, they use a set square. They slide it along the wall until the other arm touches the point where they want the perpendicular.

Exam Trick

Remember: Keep one arm of the set square on the line and slide it until the other arm passes through point P. This gives you the perpendicular line instantly.

Points To Remember

Place one arm of the right angle on the line XY.
Slide the set square so the other arm touches point P.
Draw a line along the second arm passing through P.
This line is perpendicular to line XY.

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