Practical Geometry Application: Ex 14.5 Constructions

What is a Perpendicular from an External Point?

In geometry, when two lines meet at a right angle (90°), they are called perpendicular lines. Think of the corner of a book or the letter 'L'. Now, imagine a straight line drawn on a paper and a single dot somewhere above or below that line. This dot is an 'external point' because it is outside the line. The challenge of the practical geometry application ex 14 5 is to draw a straight line from this dot that meets the original line at exactly 90°. This new line is the 'perpendicular from an external point'. An interesting fact is that this perpendicular line represents the shortest possible distance from the point to the line. Any other line you draw from the point to the line will be longer!

How to Construct a Perpendicular from a Point Not on a Line

1. Step 1: Given a Line and a Point — Start by drawing a line, let's call it 'l', using your ruler. Mark a point 'P' anywhere outside this line.
2. Step 2: Draw an Arc from the Point — Place the metal tip of your compass on point P. Open the compass to a suitable width and draw a large arc that cuts the line 'l' at two distinct points. Label these points 'A' and 'B'.
3. Step 3: Draw Arcs from the Intersection Points — Now, with A as the center, and a radius more than half the length of AB, draw an arc on the side of the line opposite to point P.
4. Step 4: Create the Intersecting Arc — Without changing the compass radius, move the compass tip to point B. Draw another arc that cuts the arc you just drew in Step 3. Label this new intersection point 'Q'.
5. Step 5: Join the Points to Form the Perpendicular — Use your ruler to draw a straight line connecting point P and point Q. The line segment PQ is the required perpendicular to the line 'l' from the external point P. You can check with a protractor that it forms a 90° angle with line 'l'.

Tips for Perfect Constructions

To score full marks in geometry constructions, precision is key!

1. Use a Sharp Pencil: A blunt pencil creates thick lines, which leads to inaccurate measurements and intersections. Always use a sharp, dark pencil for your compass and ruler work.
2. Don't Change the Compass Radius: In Step 4 of the construction, it is crucial that you use the same compass opening as you used in Step 3. Changing it will result in a line that is not perpendicular.
3. Show Construction Arcs: Do not erase the arcs you draw. These arcs are proof of your method. Make them light but visible.
4. Label Everything: Clearly label all your points (like P, A, B, Q) and lines (l). This makes your construction easy to follow and understand.

Practice Questions with Solutions

• Q: Draw a line segment AB of length 7 cm. Mark a point M outside it. Using a ruler and compass, construct a perpendicular from M to the line segment AB. A: Step 1: Draw a line segment AB = 7 cm using a ruler. Mark a point M anywhere above or below AB. Step 2: Place the compass point at M and draw an arc that intersects AB at two points, say X and Y. Step 3: With X as the center and a radius more than half of XY, draw an arc on the other side of AB. Step 4: With Y as the center and the same radius, draw another arc to intersect the previous arc at a point, say N. Step 5: Join M and N. The line MN is the perpendicular to the line segment AB from point M. Final answer: The line segment MN is the required perpendicular.
• Q: Draw a horizontal line 'm'. Take a point 'K' about 4 cm below the line. Construct a line through K which is perpendicular to 'm'. A: Step 1: Use a ruler to draw a horizontal line and label it 'm'. Mark a point 'K' approximately 4 cm below it. Step 2: With K as the center, use a compass to draw an arc that cuts line 'm' at two points, say P and Q. Step 3: With P as the center and a radius greater than half the distance PQ, draw an arc above line 'm'. Step 4: With Q as the center and the same radius, draw another arc that intersects the arc from Step 3. Label the intersection point L. Step 5: Join points K and L. The line KL is perpendicular to line 'm'. Final answer: The line KL is the constructed perpendicular.
• Q: Imagine you are given a line 'p' and a point 'R' not on it. What is the first major step you would take with a compass to begin constructing a perpendicular from R to p? A: Step 1: The very first step is to set the compass to an appropriate width. Step 2: Place the metal point of the compass on the external point 'R'. Step 3: Draw a large arc that cuts the given line 'p' at two distinct points. This creates the base points for the rest of the construction. Final answer: The first step is to place the compass on point R and draw an arc to cut line 'p' in two places.
• Q: Draw a line segment CD = 10 cm. Take a point O anywhere above it. Construct a perpendicular from O to CD. Verify your construction using a protractor. A: Step 1: Draw the line segment CD of 10 cm. Mark a point O above it. Step 2: Place the compass needle at O. Draw an arc to cut CD at two points, E and F. Step 3: With E as the center and a radius more than half of EF, draw an arc below CD. Step 4: With F as the center and the same radius, draw another arc to cut the previous arc at point G. Step 5: Join OG. The line OG is the perpendicular to CD. Step 6: To verify, place the center of a protractor at the point where OG meets CD. Align the base of the protractor with CD. The line OG should align with the 90° mark. Final answer: The line OG is the perpendicular, and it should measure 90° on a protractor.

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