A rectangle also is a special type of
parallelogram and hence all properties of a parallelogram apply to rectangles
also. A rectan-gle is a parallelogram in which each of the angles is equal to
90°. The diagonals of a rectangle are equal (and bisect each other) .
Area of a rectangle = Length x Breadth
When a rectangle is inscribed in a circle,
the diagonals become the diameters of the circle. .
A square is a rectangle in which all four
sides are equal (or a rhombus in which all four angles are equal, i.e., all are
right angles) Hence, the diagonals are equal and they bisect at right angles. .
Area of a Square = Side x Side = Side2
and Diagonal = x side.
When a square is inscribed in a circle, the
diagonals become the diameters of the circle. .
When a circle is inscribed in a square, the
side of the square is equal to the diameter of the circle. .
Any figure with three or more sides is called
a polygon. A regular polygon is one in which all sides are equal and all angles
are equal. A regular polygon can be inscribed in a circle. .
The names of polygons with three, four, five,
six, seven, eight, nine and ten sides are respectively triangle, quadrilateral,
pentagon, hexagon, heptagon, octagon, nonagon, decagon. .
The sum of interior angles of a convex
polygon is equal to (2n - 4) right angles where n is the number of the sides of
the polygon. .
A convex polygon is the one, in which each of
the interior angles is less than 180°. It can be noticed that any straight line
drawn cutting a convex polygon passes only two sides of the polygon where as in
a concave polygon, it is possible to draw lines passing through more than two
If each of the sides of a convex polygon is
extended, the sum of the external angles thus formed is equal to 4 right angles
(i.e., 360°) .
In a regular polygon of n sides, if each of
the interior angles is dE. .
d = x 90°
Each exterior angle =
Area of a regular polygon = 1/2 x Perimeter x
Perpendicular distance from the centre to any side. .
Some more useful points about quadrilaterals and triangles
- A rectangle and
a parallelogram lying on the same base and between the same parallels are equal
in area. .
- If a rectangle
(or a parallelogram) and a triangle are on the same base and between the same
parallels, then the area of the triangle will be equal to half the area of the
rectangle (or parallelogram). .
- Two triangles
with the same base and lying between the same parallels are equal in area. .
- A circle is
symmetric about any diameter. .
- Circumference of
the circle is 2pr (=pd) and area of the circle is pr2
- A chord is a
line joining two points on the circumference of a circle
- A secant is a
line intersecting a circle in two distinct points. .
- If PAB and PCD
are two secants, then PA.PB = PC.PD (not with reference to the figure given) .
- A line that
touches the circle at only one point is a tangent to the circle (RR is a
tangent touching the circle at R). .
- A tangent is
perpendicular to the radius drawn at therpcInt of tangency (RR ^ OR) i.e. at R. .
- Two tangents can
be drawn to the circle from any oint outside the circle and these two tangents
are equal in length (X is the external point and the two tangents XY and XY’
are equal.) .
- A perpendicular
drawn from the centre of the circle to a chord bisects the chord (OA, the
perpendicular from 0 to PQ bisects PQ) and conversely, the perpendicular
bisector of a chord passes through the centre of the circle. .
- Two chords that
are equal in length will be equidistant from the centre, and conversely two
chords which have their mid-points equidistant from the centre of the circle,
will be of the same length.(not with reference to the figure given) .
- The angle
between a tangent and a chord through the point of contact of the tangent is
equal to the angle made by the chord in the alternate segment (i.e., segment of
the circle other than the side of location of the angle between the tangent and
the chord) (not with reference to the figure given) .
- One and only one
circle passes through any three given non-collinear points. .
- Two circles are
said to touch each other if a common tangent can be drawn touching both the
circles at the same point. This is called the point of contact of the two
circles. When two circles touch each other, then the point of contact and the
centres of the two circles are collinear. .
- If two circles
touch internally, the distance between the two centres is equal to the
difference in the radii of the two circles. .
- A tangent drawn
common to two circles is called a direct common tangent if the tangent cuts the
line joining the centres not between the two circles but on one side of the
- A common tangent
that cuts the line joining the centres in between the two circles is called
transverse common tangent. In general, for two non-intersecting and
non-enclosed circles, two direct and two transverse common tangents can be
- Two intersecting
circles do not have a transverse common tangent. .
- Two circles
touching each other externally have two direct common tangents but only one
transverse common tangent at the point of contact. .
- Two circles
touching each other internally have totally only one common tangent at the
point of contact.
- If one circle is
completely enclosed in another circle no common tangent is possible for these
two circles. .
- Two circles are
said to be concentric if they have the same centre. As is obvious, here the
circle with smaller radius lies completely within the circle with bigger radius.
Arcs and Sectors
An are is a segment of a circle. In the
figure, ACB is called minor are and ADB is called major arc. In general, if we
talk of an are AB, we refer to the minor arc. AOB is called the angle formed by
the are (at the centre of the circle). .
- The angle
subtended by an are at the centre is double the angle subtended by the are in
the remaining part of the circle. In the figure, ∠AOB = 2 x ∠AXB. .
- Angles in the
same segment are equal. In the figure, ∠AXB = ∠AYB. .
- As we have
already seen in quadrilaterals, the opposite angles of a cyclic quadrilateral
are supplementary. It follows that the external angle of a cyclic quadrilateral
is equal to the interior opposite angle. .
The angle inscribed in a semicircle (or the
angle the diameter subtends in a semicircle) is a right angle. The converse of
the above is also true and is very useful in a number of cases - in a right
angled triangle, semi-circle can be drawn passing through the third vertex with
the hypotenuse as the diameter. .
If q is the angle made by an are ALI at the centre of the circle, then
length of are AB = 2pr x q/360. The arca formed by an arc and the two radii at the two end
points of the arc is called sector. In the adjacent figure, AOB is a sector. If
∠AOB = q, then Area of sector AOB = x pr2 x qr2