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

We will find two points on the line that are equally far from P. Then we will draw the perpendicular bisector of the segment connecting these two points. Step 1 — Mark two points on line Let us draw a line . Let us mark a point P outside line . Place the compass needle on point P. Open the compass to a suitable radius. The radius must be large enough to cross line . Draw an arc that cuts line at two distinct points. Let us name these points A and B. Step 2 — Draw intersecting arcs from A and B Now, place the compass needle on point A. Open the compass to a radius. This radius must be more than half the length of segment AB. Draw an arc above line . Draw another arc below line . Keep the compass opening exactly the same. Place the compass needle on point B. Draw an arc above line . This arc should intersect the first arc. Draw another arc below line . This arc should intersect the second arc. Let us name the points where these arcs intersect C and D. Step 3 — Draw the perpendicular line Use a ruler to connect points C and D. Draw a straight line passing through C and D. Observe that this line also passes through point P. This line is perpendicular to line . Let us call this new line m. > Answer > (i) A line m has been constructed. > (ii) Line m passes through point P. > (iii) Line m is perpendicular to line l.

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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

We will find two points on the line that are equally far from P. Then we will draw the perpendicular bisector of the segment connecting these two points.

Step 1 — Mark two points on line $l$

Let us draw a line $l$ . Let us mark a point P outside line $l$ .

Place the compass needle on point P. Open the compass to a suitable radius. The radius must be large enough to cross line $l$ . Draw an arc that cuts line $l$ at two distinct points. Let us name these points A and B.

Diagram 1

Step 2 — Draw intersecting arcs from A and B

Now, place the compass needle on point A. Open the compass to a radius. This radius must be more than half the length of segment AB. Draw an arc above line $l$ . Draw another arc below line $l$ . Keep the compass opening exactly the same. Place the compass needle on point B. Draw an arc above line $l$ . This arc should intersect the first arc. Draw another arc below line $l$ . This arc should intersect the second arc. Let us name the points where these arcs intersect C and D.

Diagram 2

Step 3 — Draw the perpendicular line

Use a ruler to connect points C and D. Draw a straight line passing through C and D. Observe that this line also passes through point P. This line is perpendicular to line $l$ . Let us call this new line m.

$\boxed{\text{Line m is perpendicular to line l and passes through P.}}$

Diagram 3

Answer

(i) A line m has been constructed. (ii) Line m passes through point P. (iii) Line m is perpendicular to line l.

Question 22

Answer

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