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# 10th Class Chapter No 18 - Circle : Constructions in Maths for ICSE

#### Constructions

The drawing of various shapes using only a pair of compasses and straightedge or ruler. No measurement of lengths or angles is allowed.

#### Constructing a tangent to a circle through a point on its circumference.

• Let O be the centre of the given circle and P be any point on its circumference
• Join O and P
• Draw a line TP making an angle of 9090∘ with OP i.e., OPT = 9090∘
• PT is the required tangent

#### Constructing a tangent to a circle from an external point

• Let O be the centre of the given circle and P be any external point
• Join O and P
• Draw a circle with OP as diameter which cuts the given circle at X and Y
• Join PX and PY
• PX and PY are the tangents to the given circle from the external point P

#### Constructing circumcircle of a triangle

• Draw the perpendicular bisectors of any two sides say AB and BC of the triangle
• Let the perpendicular bisectors of AB and BC meet at point O
• Taking O as centre and radius equal to OA (or OB or OC) draw a circle which will pass through all the vertices
• This circle is the circumcircle of the triangle

#### Constructing incircle of a triangle

• Draw the bisectors of any two angles of the triangle
• Let the bisectors meet at a point I
• From I, draw a perpendicular line to any side of the triangle
• With I as centre and the perpendicular as radius, draw a circle which will touch all the sides of the triangle
• This circle is the inscribed circle of a triangle

#### Construct a circle circumscribing a given regular hexagon

A circumscribed polygon is a polygon whose vertices are outside the edge of a circle and have every edge of the polygon touch the circle exactly once.
1. Draw the perpendicular bisectors of any two sides
2. Taking the point of intersection of these perpendicular bisectors as centre and its distance from any vertex of the given regular hexagon as radius, draw a circle

This circle is the circumscribing circle

#### Construct an inscribing circle of a given regular hexagon

A regular hexagon is a six-sided figure in which all of its angles are congruent and all of its sides are congruent.
STEPS:
1. Place your compass point on the paper and draw a circle. (Keep this compass span!)

2. Place a dot, labeled P, anywhere on the circumference of the circle to act as a starting point.

3. Without changing the span on the compass, place the compass point on P and swing a small arc crossing the circumference of the circle.

4. Without changing the span on the compass, move the compass point to the intersection of the previous arc and the circumference and make another small arc on the circumference of the circle.

5. Keep repeating this process of "stepping" around the circle until you return to point P.

6. Starting at P, connect to each arc on the circle forming the regular hexagon.

Posted in 10th on July 11 2020 at 07:13 AM

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