Subscribe to our ▶️ YouTube channel 🔴 for the latest videos, updates, and tips.
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.
Construction of Perpendicular Lines by using a Protractor.
Sum of Angles of a Quadrilateral.
Practice Test on Quadrilaterals.
5th Grade Geometry
5th Grade Math Problems
From Construction of Perpendicular Lines by Using a Protractor to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
