IM Commentary This task presents a foundational result in geometry, presented with deliberately sparse guidance in order to allow a wide variety of approaches. Teachers should of course feel free to provide additional scaffolding to encourage solutions or thinking in one particular direction.
The circle is a highly symmetric shape: Its symmetry group is the orthogonal group O 2,R. The group of rotations alone is the circle group T. All circles are similar. A circle's circumference and radius are proportional. The area enclosed and the square of its radius are proportional.
The circle that is centred at the origin with radius 1 is called the unit circle. Thought of as a great circle of the unit sphereit becomes the Riemannian circle.
Through any three points, not all on the same line, there lies a unique circle. In Cartesian coordinatesit is possible to give explicit formulae for the coordinates of the centre of the circle and the radius in terms of the coordinates of the three given points. Chord Chords are equidistant from the centre of a circle if and only if they are equal in length.
The perpendicular bisector of a chord passes through the centre of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector are: A perpendicular line from the centre of a circle bisects the chord.
The line segment through the centre Circle and tangent radius a chord is perpendicular to the chord. If a central angle and an inscribed angle of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle.
If two angles are inscribed on the same chord and on the same side of the chord, then they are equal. If two angles are inscribed on the same chord and on opposite sides of the chord, then they are supplementary.
For a cyclic quadrilateralthe exterior angle is equal to the interior opposite angle. An inscribed angle subtended by a diameter is a right angle see Thales' theorem.
The diameter is the longest chord of the circle. Among all the circles with a chord AB in common, the circle with minimal radius is the one with diameter AB.
A line drawn perpendicular to a tangent through the point of contact with a circle passes through the centre of the circle. Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal in length. Corollary of the chord theorem.
A tangent can be considered a limiting case of a secant whose ends are coincident. The angle between a chord and the tangent at one of its endpoints is equal to one half the angle subtended at the centre of the circle, on the opposite side of the chord Tangent Chord Angle.
If two secants are inscribed in the circle as shown at right, then the measurement of angle A is equal to one half the difference of the measurements of the enclosed arcs DE and BC.
This is the secant-secant theorem. Inscribed angles See also: Inscribed angle theorem Inscribed angle theorem An inscribed angle examples are the blue and green angles in the figure is exactly half the corresponding central angle red.
Hence, all inscribed angles that subtend the same arc pink are equal. Angles inscribed on the arc brown are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle since the central angle is degrees.
Sagitta The sagitta is the vertical segment. The sagitta also known as the versine is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the arc of the circle.
Given the length y of a chord, and the length x of the sagitta, the Pythagorean theorem can be used to calculate the radius of the unique circle that will fit around the two lines:tangent radius right angle point outside the circle perpendicular When we have a tangent to a circle which means it's a line or line segment or even a ray that intersects the circle on only one point if we had to draw a radius to .
Circle Theorem 7 link to dynamic page Previous Next > Alternate segment theorem: The angle (α) between the tangent and the chord at the point of contact .
The tangent intersects the circle's radius at a 90° angle; Since a tangent only touches the circle at exactly one and only one point, that point must be perpendicular to a radius. To test out the interconnected relationship of these two defining traits of a tangent, try the interactive applet.
A line drawn perpendicular to a radius through the end point of the radius lying on the circle is a tangent to the circle. A line drawn perpendicular to a tangent through the point of contact with a circle passes through the centre of the circle. Two of these four solutions give tangent lines, as illustrated above, and the lengths of these lines are equal (Casey , p.
29). The "Unit Circle" is a circle with a radius of 1. Being so simple, it is a great way to learn and talk about lengths and angles.
The center is put on a graph where the x axis and y axis cross, so we get this neat arrangement here. Have a try!
Move the mouse around to see how different angles (in.