Construction Full Video || Chapter – 11 || Class 10th (For Hindi Medium)

__Dividing a Line Segment__

**Bisecting a Line Segment**

**Step 1**: With a radius of more than half the length of the line-segment, draw arcs centred at either **end** of the line segment so that they intersect on either **side** 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 AX, making an acute angle with line segment AB.

**Step 2**: Locate 5 (= m + n) points A1,A2,A3,A4andA5 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.

Division of a line segment

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**Constructing Similar Triangles**

**Constructing a Similar Triangle with a scale factor**

Suppose we want to construct a triangle whose sides are 3/4 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.

The same procedure can be followed when the scale factor > 1.

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**Drawing Tangents to a Circle**

**Tangents: Definition**

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.

PQ is the tangent, touching the circle at A

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**Number of Tangents to a circle from a given point**

i) If the point in an **interior region of the circle**, any line through that point will be a secant. So, in this case, there is no tangent to the circle.

AB is a secant drawn through the point S

ii) When the point lies on the circle, there is accurately only one tangent to a circle.

PQ is the tangent touching the circle at A

iii) When the point lies outside of the circle, there are **exactly two tangents** to a circle.

PT1 and PT2 are tangents touching the circle at T1 and T2

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 centre O and let P be the exterior point from which the tangents to be drawn.

**Step 1**: Join the PO and bisect it. Let M be the midpoint of PO.

**Step 2**: Taking M as the 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.