Question 4
Recreate this design using only a ruler and compass —
We want to make a design using only a ruler and a compass.
Step 1 — Draw a square
First, let us draw a square. Let us name its vertices A, B, C, and D. We can choose any side length for our square.
Step 2 — Find midpoints of sides
Next, we find the middle points of each side. Let us find the midpoint of side AB. Place the compass needle at point A. Open the compass more than half of AB. Draw arcs above and below AB. Place the compass needle at point B. Use the same opening. Draw arcs that cut the first arcs. Join the two points where arcs cut. Use a ruler. This line is the perpendicular bisector of AB. It cuts AB at its midpoint. Let us call this point P. Repeat this process for side BC. Find its midpoint, call it Q. Find midpoint of CD, call it R. Find midpoint of DA, call it S. Lines PR and QS are perpendicular bisectors. They cross each other at the center of the square.
Step 3 — Draw the semicircles
Now, we will draw the curved parts. Let us set the compass opening. The radius will be the distance from A to P. This is half the length of side AB. So, the radius is AP. Place the compass needle at point P. Draw a semicircle from A to B. This semicircle will be inside the square. Repeat this for the other sides. Place the compass needle at point Q. Draw a semicircle from B to C. Place the compass needle at point R. Draw a semicircle from C to D. Place the compass needle at point S. Draw a semicircle from D to A.
Step 4 — Colour the boundary
Finally, we make our design stand out. We use a coloured pencil. Colour the boundary of the shape. This means colouring the four semicircles. This will highlight the design.
Answer
The design is created by following these steps:
(i) Draw a square and label its vertices A, B, C, D. (ii) Find the midpoints of the sides AB, BC, CD, and DA using perpendicular bisectors. Label these midpoints P, Q, R, and S respectively. (iii) Set the compass radius to the length of AP (half the side length of the square). (iv) With P as the center, draw a semicircle connecting points A and B inside the square. (v) With Q as the center, draw a semicircle connecting points B and C inside the square. (vi) With R as the center, draw a semicircle connecting points C and D inside the square. (vii) With S as the center, draw a semicircle connecting points D and A inside the square. (viii) Colour the boundary of the resulting shape to make the design visible.
More questions in FIO
When constructing the perpendicular bisector, is it necessary to have the same radius for the arcs above and below XY? Explore this through construction, and then justify your answer.
[Hint 1: Any point that is of the same distance from X and Y lies on the perpendicular bisector.
Hint 2: We can draw the whole line if any two of its points are known.]
Is it necessary to construct the pairs of arcs above and below XY? Instead, can we construct both the pairs of arcs on the same side of XY? Explore this through construction, and then justify your answer.
While constructing one pair of intersecting arcs, is it necessary that we use the same radii for both of them ? Explore this through construction, and then justify your answer.
Recreate this design using only a ruler and compass —
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[Hint: Find the angles in this figure.]
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[Hint: Find a line segment on whose perpendicular bisector passes through P.]
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