Practical Geometry Application: Ex 14.1 (Circles)

Understanding the Circle and Its Parts

Before we start drawing, let's understand what a circle is. Imagine you have a fixed point, like a pole in a playground. Now, imagine walking around that pole, always keeping the exact same distance from it. The path you walk will form a perfect circle!

So, a circle is a collection of all points that are at a fixed distance from a fixed point.

• Center: The fixed point is called the center of the circle. We usually name it with a capital letter, like O or C.
• Radius: The fixed distance from the center to any point on the circle is called the radius. If the radius is 5 cm, every single point on the circle's edge is exactly 5 cm away from the center. We often use the letter 'r' for radius.
• Diameter: A line segment that passes through the center and has its endpoints on the circle is called the diameter. It is always twice the length of the radius (Diameter = 2 × Radius).

Step-by-Step: How to Draw a Circle with a Compass

1. Step 1: Mark the Center — Take a sharpened pencil and mark a point on your paper. This point will be the center of your circle. Let's call it point O.
2. Step 2: Measure the Radius — Decide the radius you want. For example, let's choose 4 cm. Use your ruler to measure 4 cm. Place the metal tip of your compass on the '0' mark of the ruler and open the compass until the pencil tip is exactly on the '4 cm' mark.
3. Step 3: Place the Compass — Be careful not to change the compass width you just set! Now, place the metal tip of the compass firmly on the center point O that you marked on your paper.
4. Step 4: Draw the Circle — Hold the compass from the top knob. Gently rotate the pencil arm all the way around to draw the circle. Try to do it in one smooth motion. You have now drawn a perfect circle with a radius of 4 cm!

Worked Examples from Exercise 14.1

• Example 1: Draw a circle of radius 3.2 cm. Step 1: Use a ruler to set the distance between the two arms of the compass to 3.2 cm. Step 2: Mark a point 'O' on your paper to be the center. Step 3: Place the pointer of the compass on O. Step 4: Turn the compass slowly to draw the circle. The resulting circle has a radius of 3.2 cm.
• Example 2: With the same center O, draw two circles of radii 4 cm and 2.5 cm. Step 1: Mark a point 'O' as the center. Step 2: Use the ruler to set the compass to a 4 cm radius. Step 3: Place the compass pointer on O and draw the first circle. Step 4: Without moving the center, now use the ruler to set the compass to a 2.5 cm radius. Step 5: Place the compass pointer back on O and draw the second, smaller circle inside the first one. These are called concentric circles because they share the same center.

Tips for Perfect Circles

• Use a Sharp Pencil: A blunt pencil will create a thick, inaccurate line. Always keep the pencil in your compass sharp.
• Firm Center Point: Press the metal tip of the compass firmly enough so it doesn't slip, but not so hard that it tears the paper.
• Hold it Right: Hold the compass by the knob at the very top, not by its arms. Holding the arms can change the radius measurement while you are drawing.
• Check Your Measurement: Before you draw, double-check the compass opening against your ruler to make sure the radius is correct.

Practice Questions with Solutions

• Q: Draw a circle of radius 4.5 cm. A: Step 1: Mark a point C on your paper. This will be the center. Step 2: Take a ruler and a compass. Open the compass so that the distance between the metal tip and the pencil tip is 4.5 cm. Step 3: Place the metal tip of the compass on point C. Step 4: Rotate the pencil arm smoothly around point C to complete the circle. This is the required circle with a radius of 4.5 cm.
• Q: Draw a circle with center O and radius 4 cm. Mark points P, Q, and R such that P is on the circle, Q is in the interior of the circle, and R is in the exterior of the circle. A: Step 1: Draw a circle with center O and radius 4 cm using a ruler and compass. Step 2: To mark point P on the circle, simply choose any point on the line you just drew and label it P. Step 3: To mark point Q in the interior, choose any point inside the circle's boundary and label it Q. The distance OQ will be less than 4 cm. Step 4: To mark point R in the exterior, choose any point outside the circle's boundary and label it R. The distance OR will be greater than 4 cm. Final answer: You will have a circle with points P on the boundary, Q inside, and R outside.
• Q: Draw a circle and any two of its diameters. If you join the ends of these diameters, what is the figure obtained? A: Step 1: Draw a circle of any radius with center O. Step 2: Draw a line segment AB that passes through the center O and has its endpoints A and B on the circle. This is one diameter. Step 3: Draw another line segment CD that also passes through the center O and has its endpoints C and D on the circle. This is the second diameter. Step 4: Join the endpoints in order: A to C, C to B, B to D, and D to A. Final answer: The figure ACBD obtained is a rectangle. If the diameters are perpendicular, the figure is a square.
• Q: Draw a circle of radius 3.5 cm. Draw any chord in it. Construct the perpendicular bisector of the chord and see if it passes through the center. A: Step 1: Draw a circle with a radius of 3.5 cm and center O. Step 2: Draw any line segment with endpoints on the circle, for example, AB. This is a chord. Step 3: To construct the perpendicular bisector of AB, place the compass pointer on A and, with a radius more than half of AB, draw arcs above and below AB. Step 4: Keeping the same compass radius, place the pointer on B and draw arcs to cut the previous arcs at points X and Y. Step 5: Join X and Y with a straight line. This line is the perpendicular bisector of the chord AB. Final answer: You will observe that the line XY passes through the center O of the circle. This is a property of all circles.

Frequently Asked Questions