Dividing a Line Segment Division of a Line Segment 1) Bisecting a Line Segment Step 1 : With a radius more than half the length of the line-segment, draw arcs centred at either  ends  of the line segment so that they intersect on either  sides  of the line segment. Step 2 : Join the points of intersection. The line segment is bisected by the line segment joining the points of intersection. PQ is the perpendicular bisector of AB 2) Given a line segment AB, divide it in the ratio  m:n , where both m and n are positive integers. Suppose we want to divide AB in the ratio 3:2 (m=3, n=2) Step 1 : Draw any ray  → A X , making an acute angle with  ¯ ¯¯¯¯¯¯ ¯ A B .  Step 2 : Locate 5 (= m + n) points  A 1 , A 2 , A 3 , A 4   a n d   A 5  on AX such that  A A 1 = A 1 A 2 = A 2 A 3 = A 3 A 4 = A 4 A 5 Step 3 : Join  B A 5 . ( A ( m + n ) = A 5 ) Step 4 : Through the point  A 3 ( m = 3 ) , draw a line parallel to  B A 5  (by making an angle equal to  ∠ A A 5 B )  at  A

Introduction to Circles Circle and line in a plane For a circle and a line on a plane, there can be  three  possibilities.  i) they can be  non-intersecting ii) they can have  a single common point  - in this case, the line touches the circle. ii) they can have  two common points  - in this case, the line cuts the circle. (i) Non intersecting   (ii) Touching  (iii) Intersecting Tangent A  tangent to a circle  is a line which touches the circle at exactly one point. For every point on the circle, there is a unique tangent passing through it. Tangent Secant A  secant to a circle  is a line which has two points in common with the circle. It cuts the circle at two points, forming a chord of the circle. Secant Tangent as a special case of Secant Tangent as a special case of Secant The tangent to a circle can be seen as a special case of the secant, when the two end points of its corresponding chord coincide. Two parallel tangents at most for a given secant

eights and Distances Horizontal Level and Line of Sight Line of sight and horizontal level Line of sight  is the line drawn from the eye of the observer to the point on the object viewed by the observer. Horizontal level  is the horizontal line through the eye of the observer. Angle of elevation The  angle of elevation  is relevant for objects above horizontal level.  It is the  angle  formed by the  line of sight  with the  horizontal level . Angle of elevation Angle of depression The  angle of depression  is relevant for objects below horizontal level. It is the  angle  formed by the  line of sight  with the  horizontal level . Angle of depression Calculating Heights and Distances To, calculate heights and distances, we can make use of trigonometric ratios. Step 1 : Draw a  line diagram  corresponding to the problem. Step 2 : Mark all known heights, distances and angles and denote unknown lengths by variables. Step 3 : Use the values of various  trigon