Draw a line and mark a point anywhere outside the line. Construct a perpendicular to the given line through . [Hint: Find a line segment on whose perpendicular bisector passes through .]

Understand the Question To draw a perpendicular line to line passing through an external point , we first locate two points and on line that are equidistant from . The line segment then has its perpendicular bisector passing directly through . By constructing the perpendicular bisector of segment , we obtain the required line perpendicular to through . Step 1 · Mark two points on line Draw a straight line and mark a point outside it. Place the compass point at and open it to a convenient radius large enough to intersect line . Draw an arc cutting line at two distinct points, and . Step 2 · Draw intersecting arcs from and With as the center and a radius greater than , draw arcs on both sides of line . Using the same radius and with as the center, draw arcs intersecting the previous arcs at points and . Step 3 · Draw the perpendicular line Join points and with a straight line . The line passes through point and is perpendicular to line . Answer > Line is the required perpendicular line to line passing through point . Common Mistakes Radius Too Small from : If the compass radius chosen from is shorter than the distance from to line , the arc will not intersect line at two points. Radius Less than : When drawing intersecting arcs from and , setting the radius to less than half the length of will prevent the arcs from intersecting. Changing Compass Width: Changing the compass radius between drawing the arc from and the arc from will result in a line that is not perpendicular to .

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Question 22

Draw a line $l$ and mark a point $P$ anywhere outside the line. Construct a perpendicular to the given line $l$ through $P$ .

[Hint: Find a line segment on $l$ whose perpendicular bisector passes through $P$ .]

Solution

Understand the Question

To draw a perpendicular line to line $l$ passing through an external point $P$ , we first locate two points $A$ and $B$ on line $l$ that are equidistant from $P$ .
The line segment $AB$ then has its perpendicular bisector passing directly through $P$ .
By constructing the perpendicular bisector of segment $AB$ , we obtain the required line perpendicular to $l$ through $P$ .

Step 1 · Mark two points on line $l$

Draw a straight line $l$ and mark a point $P$ outside it. Diagram 1

Place the compass point at $P$ and open it to a convenient radius large enough to intersect line $l$ .
Draw an arc cutting line $l$ at two distinct points, $A$ and $B$ .

Step 2 · Draw intersecting arcs from $A$ and $B$

Diagram 2

With $A$ as the center and a radius greater than $\dfrac{1}{2}AB$ , draw arcs on both sides of line $l$ .
Using the same radius and with $B$ as the center, draw arcs intersecting the previous arcs at points $C$ and $D$ .

Step 3 · Draw the perpendicular line

Diagram 3

Join points $C$ and $D$ with a straight line $m$ .
The line $m$ passes through point $P$ and is perpendicular to line $l$ .

Answer

Line $m$ is the required perpendicular line to line $l$ passing through point $P$ .

Common Mistakes

Radius Too Small from $P$ : If the compass radius chosen from $P$ is shorter than the distance from $P$ to line $l$ , the arc will not intersect line $l$ at two points.
Radius Less than $\frac{1}{2}AB$ : When drawing intersecting arcs from $A$ and $B$ , setting the radius to less than half the length of $AB$ will prevent the arcs from intersecting.
Changing Compass Width: Changing the compass radius between drawing the arc from $A$ and the arc from $B$ will result in a line that is not perpendicular to $l$ .

Question 22

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