Question 16
Construct the following figure.
We will construct the 8-pointed star by first drawing a circle and marking 8 equally spaced points. Then we will connect these points to form the star's outline. Finally, we will shade the correct regions.
Step 1 — Draw the base circle and points
First, we use a compass to draw a circle. Let us mark the center of this circle as point O. Next, we draw a horizontal diameter and a vertical diameter through point O. These diameters will give us 4 equally spaced points on the circle. Now, we bisect the angles between these diameters. We draw two more diameters along these bisectors. These new diameters will give us 4 more points on the circle. In total, we now have 8 equally spaced points on the circle. Let us label these points S, T, U, V, W, X, Y, Z in clockwise order, starting from the top point as S.
Step 2 — Draw the star outline
Now, we will connect the points on the circle to form the star. We draw a straight line segment from point S to point V. Then, we draw a straight line segment from point T to point W. Next, we draw a straight line segment from point U to point X. We continue this pattern by drawing a straight line segment from point V to point Y. Then, we draw a straight line segment from point W to point Z. Next, we draw a straight line segment from point X to point S. We draw a straight line segment from point Y to point T. Finally, we draw a straight line segment from point Z to point U. These eight line segments form the complete outline of the 8-pointed star.
Step 3 — Identify and shade the regions
The star is now fully drawn. We can see there are 8 outer triangular regions, which are the "points" of the star. These regions are labeled A, B, C, D, E, F, G, H in the original diagram. We leave these 8 outer triangular regions unshaded (white). The remaining triangular regions are located between these points, forming the "body" of the star. These are the regions that are shaded in the original diagram. Carefully shade these 8 inner triangular regions to match the given figure.
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 —
Justify why AB in Fig. 6.4 is the perpendicular bisector.
Can you think of different methods to construct a 90° angle at a given point on a line using a rope?
Construct at least 4 different angles. Draw their bisectors.
Construct the 8-petalled figure shown in Fig. 6.5.
In Step 2 of angle bisection, if arcs of equal radius are drawn on the other side, as shown in the figure, will the line OC still be an angle bisector? Explore this through construction, and then justify your answer.
What are the other angles that can be constructed using angle bisection? Can you construct 65.5° angle?
Come up with a method to construct the angle bisector using a rope.
Construct the following figure.
How do we construct the petals so that they are of the maximum possible size within a given square?
Construct at least 4 different angles in different orientations without taking any measurement. Make a copy of all these angles.
Construct the Fig. 6.6.
Construct 4 pairs of parallel lines in different orientations.
Construct the following figure.
Use support lines in Fig. 6.11 to construct a pointed arch. Make different arches, by changing the radius of the arcs.
Make your own arch designs.
Construct the following figures:
Optical Illusion: Do you notice anything interesting about the following figure? How does this happen? Recreate this in your notebook.
Construct this figure.
[Hint: Find the angles in this figure.]
Draw a line and mark a point P anywhere outside the line. Construct a perpendicular to the given line through P.
[Hint: Find a line segment on whose perpendicular bisector passes through P.]
How can the tangram pieces be rearranged to form each of the following figures?
Are the following tilings possible?