Selina solutions for कन्साइस माठेमटिक्स [इंग्रजी] इयत्ता १० आयसीएसई chapter 19 - Constructions (Circles) [Latest edition]

In the given figure, AP is bisector of angle A of △ABC and DP is perpendicular bisector of side AB, then:

Incentre of a triangle is the point of intersection of the ______.

For a regular hexagon, inscribing a circle, the length of the side of the hexagon and the radius of the circle are ______.

For a regular hexagon of perimeter 18 cm inscribed in a circle, the radius of the circle is ______.

In triangle ABC, bisectors of angles A and B meet at point P.

Assertion (A): PC bisects angle C.

Reason (R): Bisectors of angles of a triangle are concurrent.

In triangle ABC, ∠A = 35° and ∠C = 55°.

Assertion (A): Circle with AC as diameter will pass through the vertex B. 

Reason (R): ∠ABC = 180° − 35° + 55°

= 90°

= angle of semi-circle

In ΔABC, PQ is perpendicular bisector of side AB and PR is perpendicular bisector of side BC.

Statement (1): Perpendicular bisector of side AC will pass through point P.

Statement (2): Perpendicular bisectors of the sides of a triangle are concurrent.

Pentagon ABCDE is inscribed in a circle with centre O.

Statement (1): \[\angle \mathrm{AOB} = \frac{360^{\circ}}{5}\]

Statement (2): If pentagon is regular, \[\angle \mathrm{AOB} = \frac{360^{\circ}}{5}\]

Draw a circle of radius 3 cm. Mark a point P at a distance of 5 cm from the centre of the circle drawn. Draw two tangents PA and PB to the given circle and measure the length of each tangent. 

Draw a circle of radius 5 cm. Draw two tangents to this circle so that the angle between the tangents is 45°.

Using ruler and compasses only, draw an equilateral triangle of side 4.5 cm and draw its circumscribed circle. Measure the radius of the circle.  

Using ruler and compasses only,

1. Construct triangle ABC, having given BC = 7 cm, AB – AC = 1 cm and ∠ABC = 45°.
2. Inscribe a circle in the ΔABC constructed in (i) above. Measure its radius.

Using ruler and compasses only,

1. Construct a triangle ABC with the following data:
   Base AB = 6 cm, BC = 7.5 cm and angle CAB = 60°. 
2. In the same diagram, draw a circle which passes through the points A, B and C and mark its centre O. 
3. Draw a perpendicular from O to AB which meets AB in D.
4. Prove that: AD = BD.

Using ruler and compasses only construct a triangle ABC in which BC = 4 cm, ∠ACB = 45° and perpendicular from A on BC is 2.5 cm. Draw a circle circumscribing the triangle ABC and measure its radius.  

1. Using ruler and compasses only, construct a triangle ABC in which AB = 8 cm, BC = 6 cm and CA = 5 cm.
2. Find its in centre and mark it I.
3. With I as centre, draw a circle which will cut off 2 cm chords from each side of the triangle. What is the length of the radius of this circle. 

Draw a circle circumscribing a regular hexagon with side 5 cm. 

Draw an inscribing circle of a regular hexagon of side 5.8 cm.

Construct a triangle ABC in which base BC = 5.5 cm, AB = 6 cm and ∠ABC = 120°.

1. Construct a circle circumscribing the triangle ABC.
2. Draw a cyclic quadrilateral ABCD so that D is equidistant from B and C. 

Using a ruler and compasses only: 

1. Construct a triangle ABC with the following data: AB = 3.5 cm, BC = 6 cm and ∠ABC = 120°.
2. In the same diagram, draw a circle with BC as diameter. Find a point P on the circumference of the circle which is equidistant from AB and BC. 
3. Measure ∠BCP.

Draw a line AB = 5 cm. Mark a point C on AB such that AC = 3 cm. Using a ruler and a compass only, construct:

1. A circle of radius 2.5 cm, passing through A and C.
2. Construct two tangents to the circle from the external point B. Measure and record the length of the tangents.

Using a ruler and a compass, construct a triangle ABC in which AB = 7 cm, ∠CAB = 60° and AC = 5 cm. Construct the locus of :

1. points equidistant from AB and AC.
2. points equidistant from BA and BC.
   Hence construct a circle touching the three sides of the triangle internally.

Construct a triangle ABC in which AB = 5 cm, BC = 6.8 cm and median AD = 4.4 cm. Draw incircle of this triangle.

Draw two concentric circles with radii 4 cm and 6 cm. Taking a point on the outer circle, construct a pair of tangents to inner circle. By measuring the lengths of both the tangents, show that they are equal to each other.

In triangle ABC, ∠ABC = 90°, AB = 6 cm, BC = 7.2 cm and BD is perpendicular to side AC. Draw circumcircle of triangle BDC and then state the length of the radius of this circumcircle drawn.

Construct a triangle ABC with ∠ABC = 60°, AB = 6 cm and BC = 5 cm. Extend side AB to a point D so that AD = 9 cm. Also, extend side AC to a point E so that AE = 8 cm. Construct a circle which must touch AB produced, BC and AC produced.

Using ruler and compasses only:

1. Construct a triangle ABC with following data: AB = 3.5 cm, BC = 6 cm and ∠ABC = 120°.
2. In the same diagram, draw a circle with BC as diameter. Find a point P on the circumference of the circle drawn which is equidistant from A and C.