Construction of Perpendicular Lines
When two lines intersect each other to form right angles, these lines are called perpendicular to each other. Perpendicular lines lie on the same plane, which means that they are coplanar and intersect at the right angles. Therefore, it means that if you have two lines which are perpendicular to each other, these lines would be at right angles and vice versa. By using a ruler and a compass you can easily construct a line perpendicular to the other that passes through a point which lies on that line. In this article, we will learn about the construction of perpendicular lines step by step in detail.
Construction of Perpendicular Lines
Before you get into the construction procedure of the perpendicular lines, you must know about the construction of perpendicular lines in brief. The first step for constructing the line segment is that you have to draw the line segment for the given measurement. Then you need to mark the point on the line and then place the compass on that particular given point. Next, you need to draw the arc across the given line on either side. Now, without adjusting the compass, draw one more arc. You should note that when a perpendicular bisector is drawn for the given line segment, the division of the line segment would give you the equal measures on both the sides of that line.
Steps for Construction of Perpendicular Lines
Go through the following steps to draw a perpendicular line.
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The first step for the construction of perpendicular lines is placing the compass on the given point (point P according to the diagram here). Next, draw an arc across the line on both the sides of the given point. Make sure that you do not adjust the compass width when you draw the second arc.
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From both the arcs on the line, draw another arc on the opposite side of the line you have drawn from the given point P in a way that the two new arcs intersect.
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Then use your ruler for joining the given point P to the point where the arc is intersecting at the point Q.
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Hence you can say that PQ is perpendicular to the line AB.
You can also write it as PQ ⟂ AB
Constructing a Perpendicular Line Through a Point on the Line
Follow the following steps for constructing a perpendicular line through a point on the line.
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First, you need to place your compass on the point given (point P here). Then draw an arc across the line on either side of this point on the line. Make sure that you are not adjusting the compass width while you are drawing the second arc.
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The next step is opening your compass in a way that it is wider than the distance from either of the arcs to the point P. Then place the compass on either of these arcs and draw an arc above or below the given point P in a way that the two new arcs intersect.
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Next, use your ruler for joining the point P at the point Q, where the arcs are intersecting.
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Construction of Perpendicular Lines with Example
Use the diagram given below to construct a perpendicular line to the given line AB that passes through the point C.
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Solution
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First place your compass on the point Cand use any width for drawing an arc which intersects the line AB two times.
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Then, place your compass at both points A and B such that the arcs intersect. At each point, make an arc below the line AB. And lastly, label this intersection of both the arcs as D and connect D to the point C. The perpendicular line would look like this.
1. How to Draw a Perpendicular Line?
Suppose a point P is given on a line and you need to draw a perpendicular line through the given point. Follow the following steps to know how to do it.
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The first step is to place your compass on the point P and construct an arc of any size of your choice below the line such that it crosses the line two times. You need to draw at least a semicircle. When you are drawing this arc above or below the line, prefer drawing it below the arc since it keeps the construction lines from bumping into each other.
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Stretch your compass wider!
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Then place your compass on the point where the arc crosses the line on one side and makes a small arc above the given line. You can draw this arc below the line too if you wish.
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Now, do not change the span on the compass and place the compass on the point where the first arc crosses the line on the opposite side and make another arc. Make sure that the two small arcs you drew should be intersecting.
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Lastly, using a straightedge, connect the intersection point of the two small arcs you drew to point P.
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Your final perpendicular line would look like this:
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2. What is the Proof of the Construction of Perpendicular Lines?
When you construct a perpendicular line, you are actually bisecting an angle. Since the given straight line on which you are constructing a perpendicular line is at 180 degrees, you create two 80 degree angles by drawing the perpendicular. Since you are forming two different right angles, you are indirectly creating a perpendicular line.