Practical Geometry Application Ex 14.3 for Class 6 Maths

Overview of Practical Geometry Applications

Practical Geometry focuses on constructing geometric figures using only two main tools: a compass and a ruler (also known as a straightedge). A ruler helps us draw straight lines and measure lengths, while a compass allows us to draw circles or arcs of specific radii. In Exercise 14.3, you'll combine these basic actions to achieve more complex constructions. For instance, you might be asked to draw a line segment of a certain length and then find its exact midpoint without measuring, or draw a circle and then mark specific parts of it. The key is to follow a sequence of steps carefully and accurately. Each construction method has a logical reason behind it, ensuring that the resulting figure is precise. Understanding these methods is crucial for solving real-world problems that involve design and measurement. Always remember that precision is your best friend in practical geometry.

Process: Constructing a Perpendicular Bisector

1. Step 1: Draw the Line Segment — Draw a line segment, say AB, of any given length using your ruler and a sharp pencil.
2. Step 2: Open the Compass — Place the compass needle at point A. Open the compass so that its pencil tip is open more than half the length of AB. This is very important for the arcs to intersect.
3. Step 3: Draw Arcs from One End — With the compass needle at A and the same radius (more than half AB), draw an arc above and another arc below the line segment AB.
4. Step 4: Draw Arcs from the Other End — Now, without changing the compass opening, place the compass needle at point B. Draw another arc above AB and another arc below AB, ensuring they intersect the previously drawn arcs. Label the points of intersection as C and D.
5. Step 5: Draw the Perpendicular Bisector — Using your ruler, draw a straight line passing through points C and D. This line CD is the perpendicular bisector of the line segment AB. It divides AB into two equal halves and is perpendicular to AB.

Worked Example: Constructing a Circle and Bisecting its Chord

• Example: Draw a circle with a radius of 4 cm. Mark its center as O. Draw any chord PQ within the circle. Construct the perpendicular bisector of the chord PQ. Step 1: Draw the Circle Mark a point O on your paper, which will be the center of your circle. Place the compass needle at O. Open the compass so that the pencil tip is exactly 4 cm away from the needle (measure with a ruler). Rotate the compass slowly to draw a complete circle. Step 2: Draw a Chord Choose any two points on the circumference of the circle. Let's call them P and Q. Using your ruler, draw a straight line segment connecting P and Q. This is your chord PQ. Step 3: Construct Perpendicular Bisector of Chord PQ Now, treat PQ as a line segment. Place the compass needle at P. Open the compass to more than half the length of PQ. Draw an arc above and an arc below the chord PQ. Without changing the compass opening, place the compass needle at Q. Draw another arc above and another arc below PQ, ensuring they intersect the previous arcs. Label the intersection points of the arcs as R and S. Using your ruler, draw a straight line connecting R and S. This line RS is the perpendicular bisector of the chord PQ. You'll notice it passes through the center O of the circle!

Exam Tip: Precision and Clear Labeling

When performing practical geometry constructions, two things are key to scoring full marks: precision and clear labeling. Always use a sharp pencil for fine lines and accurate points. Measure carefully with your ruler and set your compass precisely. Even a slight error can lead to a figure that doesn't meet the requirements. Secondly, always label all your points (like A, B, O, P, Q) and lines (like AB, CD) clearly as you draw them. This helps you track your steps and makes it easy for your teacher to understand your construction. Unlabeled diagrams can cause confusion and may lead to loss of marks, even if the construction itself is correct.

Practice Questions with Solutions

• Q: Draw a line segment MN of length 6.5 cm. Construct its perpendicular bisector. A: Step 1: Draw a line segment MN of length 6.5 cm using a ruler. Step 2: Place the compass needle at M. Open the compass more than half the length of MN. Step 3: Draw an arc above MN and another arc below MN. Step 4: Without changing the compass opening, place the needle at N. Draw two more arcs that intersect the previous arcs. Label the intersection points as X and Y. Step 5: Draw a straight line passing through X and Y. This line XY is the perpendicular bisector of MN. Final answer: The perpendicular bisector of MN is constructed, dividing MN into two equal halves at a right angle.
• Q: Draw a circle with a radius of 4.5 cm. Mark its center as K. Draw any radius KL. A: Step 1: Mark a point K on your paper to be the center of the circle. Step 2: Open your compass to a radius of 4.5 cm, measuring with a ruler. Step 3: Place the compass needle at K and draw a complete circle. Step 4: Choose any point L on the circumference of the circle. Draw a straight line segment from K to L using a ruler. This is the radius KL. Final answer: A circle of radius 4.5 cm is drawn with center K, and a radius KL is marked.
• Q: Draw a line segment AB of length 7 cm. Using a compass, find the midpoint of AB without measuring with a ruler. A: Step 1: Draw a line segment AB of length 7 cm. Step 2: Place the compass needle at A. Open the compass to more than half the length of AB. Step 3: Draw arcs above and below AB. Step 4: Without changing the compass opening, place the needle at B. Draw arcs that intersect the first set of arcs. Label the intersection points as C and D. Step 5: Draw a straight line connecting C and D. This line intersects AB at a point. Let's call this point M. Step 6: Point M is the midpoint of AB. The perpendicular bisector construction automatically finds the midpoint. Final answer: The midpoint M of AB is found by constructing its perpendicular bisector.
• Q: Draw a circle with center P and radius 3 cm. Draw a diameter QR. What is the length of QR? A: Step 1: Mark a point P as the center of the circle. Step 2: Set the compass to a radius of 3 cm. Step 3: Draw a circle with center P and radius 3 cm. Step 4: Draw a straight line passing through the center P and touching two points on the circumference. Label these points Q and R. This is the diameter QR. Step 5: The diameter is always twice the radius. Since the radius is 3 cm, the diameter QR = 2 * 3 cm = 6 cm. Final answer: A circle with center P and radius 3 cm is drawn, and its diameter QR is 6 cm long.

Frequently Asked Questions