I’m Up and Down, and Round and Round

Jamuna has a circular piece of paper. She is trying to locate its centre. Amina gives her a suggestion. She follows the instructions and is thrilled to find that it works. Can you guess what Amina told her?

What are the rotational symmetries of a square? How many lines of reflection symmetry does it have? What about a regular pentagon? A regular hexagon?

What is the length of the longest chord in a circle of radius 5 units? Is there a smallest chord?

The locus of points at a given distance from a given point is a circle. What can we say about the locus of points equidistant from two given points? (Hint: We know that any point that is equidistant from two given points A and B lies on the perpendicular bisector of AB. Does this make the perpendicular bisector the locus? For this, we have to show that all the points on the perpendicular bisector are equidistant from A and B.)

How many circles pass through two points on a plane?

Are there circles of all possible radii passing through A and B? What is the radius of the smallest circle passing through A and B? What is the radius of the largest circle passing through A and B?

As you move away from segment AB along its perpendicular bisector, do the radii of the circles containing A and B increase or decrease?

As you go along the perpendicular bisector, will the circle drawn from that point through A and B appear more curved or less curved?

You are given two points A and B on a plane. How many squares can you draw on the same plane with A and B on the boundary? How many squares can you draw on the plane with A and B as the corners of the square?

Draw $\Delta ABC$ with $AB = 5$ cm, $\angle A = 70^\circ$ and $\angle B = 60^\circ$. Draw the circumcircle of $\Delta ABC$. Is the centre inside or outside the triangle?

Draw $\Delta ABC$ with $AB = 5$ cm, $\angle A = 100^\circ$, $AC = 4$ cm. Draw the circumcircle of $\Delta ABC$. Is the centre inside or outside the triangle?

Draw $\Delta ABC$, with $AB = 6$ cm, $BC = 7$ cm and $CA = 7$ cm. Draw the circumcircle of $\Delta ABC$. Let the circumcentre be O. Measure OA, OB, OC.

What is the least possible radius of a circle through two points A and B?

A, B and C are three collinear points. Can you find a point P such that $PA = PB = PC$? What can you say about the perpendicular bisectors of AB and BC? Draw and check. Can you show that for three collinear points A, B and C, the perpendicular bisector of AB and BC are parallel? Is it possible for a circle to pass through collinear points? Can you draw a line that cuts a given circle in three distinct points?

The circumcircle of a given $\Delta ABC$ is drawn. Can there be other triangles congruent to $\Delta ABC$ that share the same circumcircle?

Show that the triangle formed by a chord and the centre of the circle is isosceles.

Show that if two such isosceles triangles (occurring in the previous question) have equal base length, they are congruent to each other.

An isosceles triangle ABC is inscribed in a circle, with AB = AC. Show that the altitude from A to BC passes through the centre of the circle.

Two parallel chords of lengths 6 cm and 8 cm are on opposite sides of the centre of a circle. If the radius of the circle is 5 cm, find the distance between the midpoints of the chords.

Use the Baudhāyana–Pythagoras theorem to show why Theorem 6 must be true.