## Sunday, November 22, 2009

### GMAT/CAT PREPARATION EMAIL COURSE DAY 26

CIRCLES:

Circumference of a Circle = 2 x pi x radius =Pi x diameter

Area of a Circle = pi x r x r

Length of an Arc:

An arc is a piece of the circumference.
If n is the degree measure of the arc's central angle, then the formula is:
Length of an Arc = (n/360)(2 x pi x r)

Area of a Sector:

A sector is a piece of the area of a circle.
If n is the degree measure of the sector's central angle, then the formula is:
Area of a Sector = (n/360)(pi)(r)(r)

Characteristics of Circles:

Circles are closed plane curves with all points on the curve equally distant from a fixed point called the center.

A radius of a circle is a line segment from the center to any point on the circle. All radii of a circle are equal.

A chord is a line segment whose endpoints are on the circle.

A diameter of a circle is a chord that passes through the center of the circle. The diameter of a circle is twice its radius and the longest distance between two points on the circle.

A line passing through two points on a circle is a secant.

The area bounded by the circumference and an angle with vertex at the center of the circle is a sector.

An arc is a portion of a circle, usually measured in degrees.
The entire circle is 360 degrees
A semicircle (half a circle) is 180 degrees
A quarter of a circle is an arc of 90 degrees

A central angle is an angle whose vertex is the center of the circle and whose sides are radii of the circle. A central angle is equal in measure to its arc.

An inscribed angle is an angle whose vertex is on the circle and whose sides are chords of the circle. An inscribed angle is equal in measure to one-half its arc.

Properties of a Circle:

If two chords of a circle are equal, their corresponding arcs have equal measure.

Equal arcs subtend equal angles at the center.

A line from centre and perpendicular to a chord bisects the chord.

Equal chords of a circle are equidistant from the centre.

When two circles touch, their centres and their point of contact are collinear.

If the two circles touch externally, the distance between their centres is equal to sum of their radii.

If the two circles touch internally, the distance between the centres is equal to difference of their radii.

Angle at the centre made by an arc is equal to twice the angle made by the arc at any point on the remaining part of the circumference.

If two chords are equal, the arc containing the chords will also be equal.

There can be one and only one circle that touches three non-collinear points.

The angle inscribed in a semicircle is 90 degrees.