ARCS AND SECTORS
An arc is a segment of a circle. In Fig. 1.37, ACB is called minor arc and ADB is called major arc. In general, if we talk of an arc AB, we refer to the minor arc. AOB is called the angle formed by the arc AB (at the center of the circle).
The angle subtended by an arc at the center is double the angle subtended by the arc in the remaining part of the circle.
In Fig. 1.37 ,
Angles in the same segment are equal. In Fig. 1.37,
The angle between a tangent and a chord through the point of
contact of the tangent is equal to the angle made by in the chord
in the alternate segment (i.e., segment of the circle on the side
other than the side of location of the angle between the tangent
and the chord). This is normally referred to as the "alternate
segment theorem" in Fig. 1.38, PQ is a tangent to the circle at
the point T and TS is a chord drawn at the point of contact.
Considering which
is the angle between the tangent and the chord, the angle
TRS is the angle in the "alternate
segment". So,
Similarly
We have already seen in quadrilaterals, the opposite angles of a cyclic quadrilateral are supplementary and that the external angle of a cyclic quadrilateral is equal to the interior opposite angle.
The angle 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, a semi-circle with the hypotenuse as the diameter can be drawn passing through the third vertex (refer to Fig. 1.39).
The area formed by an arc and the two radii at the two end points of the arc is called sector. In Fig. 1.40, the shaded figure AOB is called the minor sector.
AREAS OF PLANE FIGURES
Menstruation is the branch of geometry that deals with the measurement of length, area and volume. We have looked at properties of plane figures till now. Here, in addition to areas of plane figures, we will also look at surface areas and volumes of "solids." Solids are objects, which have three dimensions (plane figures have only two dimensions).
Let us briefly look at the formulae for areas of various plane figures and surface areas and volumes of various solids
TRIANGLES
The area of a triangle is represented by the symbol . For any triangle, the three sides are represented by a, b, and c and the angles opposite these sides represented by A, B and C respectively
(i) For any triangle in general
(a) When the measurement of three sides a, b ,c are given
This is called Hero's formula.
(b) when base (b) and altitude (height) to that base ate given,
(c)
(d) Where R is the circumradi us of the
traingle
(e) Area=r.s where r is the inradius of the triangle and s, the semi-perimeter
Out of these five formulae, the first and the second are the most commonly used and are also more important from the examination point of view
(ii) For a right angled triangle.
(iii) For an equilateral triangle
The height of an equilateral triangle
(iv) For an isosceles triangle
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