Circles and Angles, Conjectures

Some words

A chord of a circle is a line segment whose endpoints both lie on the circle.

A secant of a circle is a line that intersects the circle at two points

An inscribed angle is the angle formed by two chords having a common endpoint. The other endpoints define the intercepted arc.

The central angle of the intercepted arc is the angle at the midpoint of the circle.

In the picture to the left, the inscribed angle is the angle ACB, and the central angle is the angle AMB.

The central angle and the inscribed angle

What’s the relation between the central angle and the inscribed angle of an arc?

Make a conjecture and write it down. It must be clearly shown from your construction that your conjecture holds. Save your construction.

Inscribed angles

What can you say about different inscribed angles from the same arc? Can you figure it out without making a construction? Make a conjecture and write it down. Argue that your conjecture holds by either referring to a construction, or by reasoning (or both)

Opposite angles of a quadrilateral in a circle

Make a circle and place four points on the circle. Make a quadrilateral using the four points. What can you say about opposite angles of the quadrilaterals?

Make a conjecture and write it down. It must be clearly shown from your construction that your conjecture holds. Save your construction.

Inscribed angle in a semi-circle

What can you say about the inscribed angles in a semi-circle? Can you figure it out without making a construction? Make a conjecture and write it down. Argue that your conjecture holds by either referring to a construction, or by reasoning (or both).

Intersecting chords

Make a circle with two intersecting chords.

Make four variables to store distances as in the picture to the left. a=AE, b=EC, c=BE and d=ED (you can use other letters or names)

If you want to store the distance between the points A and B into a variable called a, you write: a=Distance[A,B] in the input bar. You can also make a segment between the points A and B, and use the value of the segment.

Make sure you don’t use variable-names already used by the program.

Find a relation between the four distances in terms of either ratios or products.

Make a conjecture and write it down. It must be clearly shown from your construction that your conjecture holds. Save your construction.