Construction of Perpendicular Lines

Construction of perpendicular lines by using a protractor is discussed here.

Perpendicular lines are lines that intersect at right angles (90°).

The symbol '⊥' means "is perpendicular to".

For example: The corners of a room walls are perpendicular to each other.

I: Construction of a Perpendicular Line to a Line Through a Point on it:

First Method: Using Scale/Ruler and Set-Square

You are given a line, say \(\overline{AB}\). Can you draw a perpendicular line to \(\overline{AB}\) on a point P (say) of it? Follow the Working Rules given below to draw such perpendicular line.

Working Rules for Construction of Perpendicular Lines using Scale and Set-Square:

Step 1: Place a scale along the line AB as shown in the following figure.

Step 2: Place the set-square with its shortest side along AB above the scale.

Step 3: Slide the set-square firmly along \(\overleftrightarrow{AB}\) until its point Z coincides with the given point P

Step 4: Holding the set-square firmly, trace the line \(\overleftrightarrow{PZ}\) along the edge of the set-square. Thus, PZ is perpendicular to AB at P. We can write it as \(\overleftrightarrow{PZ}\) ⊥ \(\overleftrightarrow{AB}\)

Second Method: Using Scale/Ruler and Set-Square

Follow the Working Rules given on the following to draw a perpendicular line to a line segment, say AB, using ruler and compass.

Working Rules for Construction of Perpendicular Lines using Scale and Compass:

Step 1: Mark a point O on AB. With O as centre and a suitable radius, draw an arc to cut \(\overline{AB}\) at P and Q.

Step 2: With P as centre and taking a radius of more than PO, draw an arc on one side of \(\overline{AB}\).

Step 3: With Q as centre and taking the same radius, draw another arc to intersect the previous arc at R.

Step 4: The line through R and O is drawn. Now, \(\overleftrightarrow{OR}\) is the required perpendicular to \(\overline{AB}\) through O.

II: Construction of a Perpendicular Line Through a Point Not on it:

First Method: Using Scale/Ruler and Set-Squares:

Let us draw a perpendicular line to a line AB (say) from an outside point P.

Working Rules for Construction of Perpendicular Lines using Scale and Set-Square:

Step 1: Place a set-square XYZ just below the line AB in such a way that one of its sides containing right angle touches the line.

Step 2: Hold the set-square firmly and place a scale such that its edge is positioned along YZ of the set-squre.

Step 3: Holding the ruler firmly, slide the set- square along the scale until the side XY of the set square passes through the given point P.

Step 4: Keeping the set-square in this position, trace the line XY along the edge of the set- square. Thus, XY is the perpendicular line to the given line AB passing through the point P.

Second Method: Using Scale/Ruler and Compass:

Let us draw a perpendicular line to a line AB (say) through a point O located outside the line.

Working Rules for Construction of Perpendicular Lines using Scale and Compass:

Step 1: Draw \(\overline{AB}\) of any length and mark a point O outside it.

Step 2: With O as centre and a suitable radius, draw an arc to cut \(\overline{AB}\) at Land M respectively.

Step 3: With L as and taking a radius greater than \(\frac{1}{2}\)\(\overline{LM}\). draw an arc.

Step 4: With Mas centre and the same radius as in step 2, draw another arc to cut the previous arc at R.

Step 5: Join OR which intersects AB at Q.

Now, \(\overline{OQ}\) the required perpendicular from an external point O to AB.

III: Construction of a Perpendicular Line by Using a Protractor:

To construct a perpendicular to a given line ℓ at a given point A on it, we need to follow the given procedure for constructing an angle of 90° at A.

Steps of construction:

1. Let ℓ be the given line and A the given point on it.

2. Place the protractor on the line ℓ such that its base line coincides with ℓ, and its centre falls on A.

3. Mark a point B against the 90° mark on the protractor.

4. Remove the protractor and draw a line m passing through A and B.

Then line m ┴ line ℓ at A.

These are the steps to construct a perpendicular.

Solved Examples on Construction of Perpendicular Lines:

1. Draw a line segment \(\overline{AB}\) = 5 cm and mark a point C on AB such that \(\overline{AC}\) = 3 cm. Draw a perpendicular to at C by using scale/ruler and set-square.

Solution:

It is given that \(\overline{AB}\) = 5 and \(\overline{AC}\) = 3 cm

Using the above Working Rules, the required perpendicular CD is drawn on AB at C,

i.e., \(\overline{CD}\) ┴ \(\overline{AB}\).

2. Draw a line segment CD = 7.5 cm. Mark any point P outside CD. Draw a perpendicular from P to the line segment CD and measure the perpendicular distance from P.

Solution:

Steps of Construction:

Step I: Draw \(\overline{CD}\) = 7.5 cm and mark a point P outside it.

Step II: With Pas the centre and a suitable radius, draw an arc to cut \(\overline{CD}\) at L and M respectively.

Step III: With L as centre and taking a radius greater than 1/2 LM, draw an arc.

Step IV: With Mas centre and the same radius as in step 3, draw another arc to cut the previous arc at R.

Step V: Join PR which intersects CD at Q. Now, \(\overline{PQ}\) is the required perpendicular from an external point P to CD.

Also, PQ is the perpendicular distance from the external point P to the point Q on \(\overline{CD}\). Measure PQ to find its length.

Worksheet on Construction of Perpendicular Lines:

1. How many lines can be drawn which are perpendicular to a given line and pars through a given point

(i) lying on the line?

(ii) lying outside the line?

2. Draw a line segment AB. Mark any point C, on it. Through C, draw a perpendicular \(\overline{AB}\):

(i) using set-square

(ii) using compasses

3. Draw a line segment PQ = 8.5 cm and mark a point A on it such that \(\overline{PA}\) = 6.5 cm. Draw a perpendicular to \(\overline{PQ}\) A.

4. Draw a line segment CD = 8 cm. Mark any point P outside the \(\overline{CD}\). Draw a perpendicular from P to line segment CD and measure the perpendicular distance from P.

5. Draw a line segment AB = 8.5 cm. Taking a point C on \(\overline{AB}\) such that \(\overline{BC}\) = 5 cm, draw a perpendicular to AB at C.

● Quadrilaterals.

Elements of a Quadrilateral.

Types of Quadrilaterals.

Parallel Lines.

Perpendicular Lines.

Construction of Perpendicular Lines by using a Protractor.

Sum of Angles of a Quadrilateral.

Worksheet on Quadrilateral.

Practice Test on Quadrilaterals.

5th Grade Geometry

5th Grade Math Problems

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