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.
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 →AX, making an acute angle with ¯¯¯¯¯¯¯¯AB.
Step 2: Locate 5 (= m + n) points A1,A2,A3,A4 and A5 on AX such that AA1=A1A2=A2A3=A3A4=A4A5
Step 3: Join BA5.(A(m+n)=A5)
Step 4: Through the point A3(m=3), draw a line parallel to BA5 (by making an angle equal to ∠AA5B) at A3 intersecting AB at the point C.
Then, AC : CB = 3 : 2.
Constructing Similar Triangles
Constructing a Similar Triangle with a scale factor
Suppose we want to construct a triangle whose sides are 34 times the corresponding sides of a given triangle
Step 1: Draw any ray →BX making an acute angle with side ¯¯¯¯¯¯¯¯BC (on the side opposite to the vertex A).
Step 2: Mark 4 consecutive distances(since the denominator of the required ratio is 4) on ¯¯¯¯¯¯¯¯¯BX as shown.
Step 3: Join B4C as shown in the figure.
Step 4: Draw a line through B3 parallel to B4C to intersect BC at C’.
Step 5: Draw a line through C’ parallel to AC to intersect AB at A’. ΔA′BC′ is the required triangle.
Same procedure can be followed when scale factor > 1.
Drawing Tangents to a Circle
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.
No. of Tangents to a circle from a given point
i) If the point in interior region of the circle, any line through that point will be a secant. So, no tangent can be drawn to a circle passing through a point lying inside it.
ii) There is one and only one tangent to a circle passing through a point lying on the circle.
iii) There are exactly two tangents to a circle through a point lying outside the circle.
Drawing tangents to a circle from a point outside the circle
To construct the tangents to a circle from a point outside it.
Consider a circle with center O and let P be the exterior point from which the tangents to be drawn.
Step 1: Join ¯¯¯¯¯¯¯¯PO and bisect it. Let M be the midpoint of ¯¯¯¯¯¯¯¯PO.
Step 2: Taking M as centre and MO(or MP) as radius, draw a circle. Let it intersect the given circle at the points Q and R.
Step 3: Join PQ and PR
Step 3: ¯¯¯¯¯¯¯¯PQ and ¯¯¯¯¯¯¯¯PR are the required tangents to the circle.
Drawing Tangents to a circle from a point on the circle
To draw a tangent to a circle through a point on it.
Step 1: Draw the radius of the circle through the required point.
Step 2: Draw a line perpendicular to the radius through this point. This will be tangent to the circle.