Playing with Constructions | FIO

Question 2

Take a central line of a different length and try to draw the wave on it.

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Solution
Understand the Question
  • A wave pattern along a central line is formed by drawing two consecutive semicircles (half-circles) on opposite sides of the line.
  • Choose a central line segment of a specific length, for example, AB=12 cm\text{AB} = 12\text{ cm}.
  • Divide the segment at its midpoint X\text{X} into two equal segments AX=XB=6 cm\text{AX} = \text{XB} = 6\text{ cm}, each serving as the diameter of a semicircle.
  • The radius for each semicircle is half its diameter: Radius=6 cm2=3 cm\text{Radius} = \dfrac{6\text{ cm}}{2} = 3\text{ cm}.

Step 1 · Draw the Central Line

Draw a straight horizontal line segment AB\text{AB} of chosen length 12 cm12\text{ cm}.Diagram 1

Step 2 · Find the Midpoint

Mark the midpoint X\text{X} of AB\text{AB} to divide it into two equal segments AX\text{AX} and XB\text{XB}.

AX=AB2=12 cm2=6 cm\begin{aligned} \text{AX} &= \dfrac{\text{AB}}{2} \\[0.6em] &= \dfrac{12\text{ cm}}{2} \\[0.6em] &= 6\text{ cm} \end{aligned}

XB=6 cm\text{XB} = 6\text{ cm}

Step 3 · Calculate the Radius for the Semicircles

Each segment AX\text{AX} and XB\text{XB} serves as the diameter for a semicircle.

For the first semicircle on diameter AX=6 cm\text{AX} = 6\text{ cm}:

Radius1=AX2=6 cm2=3 cm\begin{aligned} \text{Radius}_1 &= \dfrac{\text{AX}}{2} \\[0.6em] &= \dfrac{6\text{ cm}}{2} \\[0.6em] &= 3\text{ cm} \end{aligned}

For the second semicircle on diameter XB=6 cm\text{XB} = 6\text{ cm}:

Radius2=XB2=6 cm2=3 cm\begin{aligned} \text{Radius}_2 &= \dfrac{\text{XB}}{2} \\[0.6em] &= \dfrac{6\text{ cm}}{2} \\[0.6em] &= 3\text{ cm} \end{aligned}

Step 4 · Draw the First Semicircle

Mark the center C1C_1 as the midpoint of AX\text{AX}. Using a compass with center C1C_1 and radius 3 cm3\text{ cm}, draw a semicircle above the line segment AX\text{AX} starting at A\text{A} and ending at X\text{X}.Diagram 3

Step 5 · Draw the Second Semicircle

Mark the center C2C_2 as the midpoint of XB\text{XB}. Using a compass with center C2C_2 and radius 3 cm3\text{ cm}, draw a semicircle below the line segment XB\text{XB} starting at X\text{X} and ending at B\text{B}.This completes the wave on the central line AB\text{AB}.

Answer

A wave is drawn on a central line AB=12 cm\text{AB} = 12\text{ cm} using two semicircles of radius 3 cm3\text{ cm} each on opposite sides of the line.

Common Mistakes
  • Same-Side Semicircles: Drawing both semicircles on the same side of the line instead of alternating (one above and one below), which fails to form a continuous wave.
  • Diameter vs. Radius Confusion: Setting the compass radius equal to the segment length (6 cm6\text{ cm}) rather than half of it (3 cm3\text{ cm}).

More questions in FIO

Q1

What radius should be taken in the compass to get this half circle? What should be the length of AXAX?

Q2

Take a central line of a different length and try to draw the wave on it.

Q3

Try to recreate the figure where the waves are smaller than a half circle (as appearing in the neck of the figure, 'A Person'). The challenge here is to get both the waves to be identical. This may be tricky!

Q4

Draw the rectangle and four squares configuration (shown in Fig. 8.3) on a dot paper.

What did you do to recreate this figure so that the four squares are placed symmetrically around the rectangle? Discuss with your classmates.

Q5

Identify if there are any squares in this collection. Use measurements if needed.

Q6

Think: Is it possible to reason out if the sides are equal or not, and if the angles are right or not without using any measuring instruments in the above figure? Can we do this by only looking at the position of corners in the dot grid?

Q7

Draw at least 3 rotated squares and rectangles on a dot grid. Draw them such that their corners are on the dots. Verify if the squares and rectangles that you have drawn satisfy their respective properties.

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