Practical Geometry Application Ex 14.4 Class 6 NCERT

What are Perpendicular Lines?

Before we begin drawing, let's understand what perpendicular lines actually are. When two lines intersect each other in such a way that the angle between them is exactly a right angle (90 degrees), they are said to be perpendicular to each other. We use the mathematical symbol $\perp$ to denote perpendicular lines. For example, if line $AB$ is perpendicular to line $CD$, we write it as $AB \perp CD$. You can see perpendicular lines all around you: the adjacent edges of your notebook, the intersection of two crossing roads, and the corner of your textbook. In Exercise 14.4, we focus on drawing a perpendicular to a given line from a specific point. There are two scenarios for this: either the point lies directly on the line, or the point lies outside the line. Mastering both methods will build a strong foundation for your higher-class geometry lessons.

Step-by-Step Construction Processes

1. Case 1: Perpendicular to a line through a point ON it (Using Ruler and Compasses) — Step 1: Draw a line segment PQ and mark a point M on it. Step 2: Place the compass pointer on M and draw an arc of any convenient radius that cuts the line PQ at two points, say A and B. Step 3: With A as center and a radius greater than AM, draw an arc above the line. Step 4: With the same radius and B as center, draw another arc cutting the previous arc at point N. Step 5: Join M and N. The line MN is the required perpendicular to PQ through point M.
2. Case 2: Perpendicular to a line from a point OUTSIDE it (Using Ruler and Compasses) — Step 1: Draw a line l and mark a point P outside it. Step 2: Place the compass pointer on point P. Draw an arc of a convenient radius that intersects the line l at two distinct points, say X and Y. Step 3: Keeping the same radius (or slightly more than half of XY), place the pointer on X and draw an arc on the opposite side of P. Step 4: Placing the pointer on Y with the same radius, draw another arc intersecting the previous arc at point Q. Step 5: Join P and Q. The line PQ is the perpendicular from point P to line l.

Pro-Tips for Perfect Constructions

• Always use a sharp pencil to ensure thin, precise lines and arcs. Thick pencil marks can lead to measurement errors.
• Ensure your compass joint is tight so the radius does not change mid-way through drawing arcs.
• When intersecting arcs, mark the point of intersection clearly with a tiny dot before drawing your straight line.
• Always verify your final angle using a protractor. If the angle is not exactly 90 degrees, check if your compass slipped.

Practice Questions with Solutions

• Q: Draw a line segment AB of length 7 cm. Mark a point P on it such that AP = 3 cm. Construct a perpendicular to AB at point P. A: Step 1: Draw a line segment AB of length 7 cm using a ruler. Step 2: From point A, measure 3 cm along AB and mark this point as P. Step 3: Place the pointer of your compass at P and draw a semi-circle arc that cuts the line AB at two points, X and Y. Step 4: Using X and Y as centers and a radius greater than PX, draw two intersecting arcs above the line segment AB. Let the point of intersection be Q. Step 5: Join point P and Q with a straight line. Final answer: Line QP is the required perpendicular to AB at point P.
• Q: Draw a line segment XY = 6 cm. Take a point M outside XY. Construct a perpendicular to XY from point M. A: Step 1: Draw a line segment XY of length 6 cm using a ruler. Step 2: Mark a point M anywhere above the line segment XY. Step 3: Place the compass point on M and swing an arc that intersects XY at two points, A and B. Step 4: With A and B as centers and a radius greater than half of AB, draw two arcs below the line segment XY that intersect each other at point N. Step 5: Place your ruler on points M and N and draw a line passing through them. Final answer: Line MN is the perpendicular to XY from point M.
• Q: Construct a line segment CD = 8 cm. Find its midpoint M. Construct a perpendicular to CD passing through M using ruler and compasses. A: Step 1: Draw a line segment CD of length 8 cm. Step 2: Mark the midpoint M at 4 cm from C using a ruler. Step 3: Place the compass point at M, draw an arc of convenient radius intersecting CD at points P and Q. Step 4: From points P and Q, draw arcs of equal radius (greater than PM) above CD. Let them intersect at point R. Step 5: Join MR. Final answer: Line MR is the perpendicular to CD at its midpoint M.
• Q: Draw a line segment PQ. Mark a point R on PQ. Construct a line RS perpendicular to PQ at R. Now construct a perpendicular to RS at point S. A: Step 1: Draw a line PQ and mark a point R on it. Step 2: Construct a perpendicular RS to PQ at R using the standard compass arc method. Step 3: Mark a point S on this newly constructed perpendicular line RS. Step 4: Treat RS as your base line. Place the compass pointer on S and construct a perpendicular to RS at point S. Step 5: Observe that the new perpendicular line is parallel to the original line PQ. Final answer: The final perpendicular drawn at S is parallel to PQ.

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