ML Aggarwal Solutions for Chapter 16 Construction Class 10 Maths ICSE
Exercise 16.1
1. Use a ruler and compass only in this question.
(i) Draw a circle, centre O and radius 4 cm.
(ii) Mark a point P such that OP = 7 cm.
Construct the two tangents to the circle from P. Measure and record the length of one of the tangents.
Answer
Steps to construct:
Step 1: Draw a circle with center O and radius 4cm and mark that point as A.
Step 2: Take a point P such that OP = 7 cm.
Step 3: Bisect OB at M.
Step 4: With center M and diameter OP, draw another circle intersecting the given circle at A and B.
Step 5: Join PA and PB. Hence PA and PB are pair of tangents to the circle.
Step 6: On measuring PA, it is equal to 5.5 cm.
2. Draw a line AB = 6 cm. Construct a circle with AB as diameter. Mark a point P at a distance of 5 cm from the mid-point of AB. Construct two tangents from P to the circle with AB as diameter. Measure the length of each tangent
Answer
Steps to construct:
Step 1: Draw a line segment AB = 6 cm.
Step 2: Draw its perpendicular bisector bisecting it at point O.
Step 3: With center O and radius OB, draw a circle.
Step 4: Extend AB to point P such that OP = 5 cm.
Step 5: Draw its perpendicular bisector intersecting it at point M.
Step 6: With center M and radius OM, draw a circle which intersects the given circle at T and S.
Step 7: Join OT, OS, PT and PS. Hence, PT and PS are the required tangents to the given circle.
Step 8: On measuring each tangent is 4cm long. PT = PS = 4 cm.
3. Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also, verify the measurement by actual calculation.
Answer
Steps to construct:
Step 1: Mark a point O.
Step 2: With center O and radius 4 cm and 6 cm, draw two concentric circles.
Step 3: Join OA and mark its mid-point as M.
Step 4: With center M and radius MA, draw another circle which intersects the first circle at P and Q.
Step 5: Join AP and AQ. Hence, AP and AQ are the required tangents to the first circle from point A.
Steps to construct:
Step 1: Consider a point O on a line, with center O, and radius 3cm, draw a circle.
Step 2: Extend its diameters on both sides and cut off OP = OQ = 7cm.
Step 3: Mark the mid-points of OP and OQ as M and N respectively.
Step 4: With Centers M and N and OP and OQ as diameters, draw circles which intersect the given circle at A, B and C, D respectively.
Step 5: Join PA, PB, QC, QD. Hence, PA, PB and QC, QD are the required tangents.
5. Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 cm and taking B as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle.
Answer
Steps to construct:
Step 1: Draw a line segment AB = 8cm.
Step 2: With center as A and radius 4cm, with center as B and radius 3cm, draw circles.
Step 3: Draw the third circle AB as diameter which intersects the given two circles at C and D, P and Q respectively.
Step 4: Join AC, AD, BP, BQ. Hence, AC and AD, BP and BQ are the required tangents.
Exercise 16.2
1. Draw an equilateral triangle of side 4 cm. Draw its circumcircle.
Steps to construct:
Step 1: Draw a line segment BC = 4cm.
Step 2: With centers B and C, draw two arcs of radius 4cm which intersects each other at point A.
Step 3: Join AB and AC.
Step 4: Draw the right bisector of BC and AC intersecting each other at point O.
Step 5: Join OA, OB and OC.
Step 6: With center as O, and radius equal to OB or OC or OA, draw a circle which passes through points A, B and C.
Hence, the required circumcircle of triangle ABC is given below.
2. Using a ruler and a pair of compasses only, construct:
(i) A triangle ABC given AB = 4 cm, BC = 6 cm and ∠ABC = 90°.
(ii) A circle which passes through the points A, B and C and mark its centre as O.
Answer
Steps to construct:
Step 1: Draw a line segment AB = 4 cm.
Step 2: At point B, draw a ray BX making an angle of 90o and cut off BC = 6 cm.
Step 3: Join AC.
Step 4: Draw the perpendicular bisectors of sides AB and AC intersecting each other at point O.
Step 5: With center as O, and radius equal to OB or OA or OC, draw a circle which passes through points A, B, C.
3. Construct a triangle with sides 3 cm, 4 cm and 5 cm. Draw its circumcircle and measure its radius.
Answer
Steps to construct:
Step 1: Draw a line segment BC = 4 cm.
Step 2: With Center as B and radius 3cm, with center as C and radius 5cm draw two arcs which intersect each other at point A.
Step 3: Join AB and AC.
Step 4: Draw the perpendicular bisector of sides BC and AC which intersects each other at point O.
Step 5: Join OB.
Step 6: With center as O and radius OB, draw a circle which pass through A, B, C.
Step 7: On measuring the radius OB = 2.5 cm.
4. Using a ruler and compasses only:
(i) Construe a triangle ABC with the following data:
Base AB = 6 cm, AC = 5.2 cm and ∠CAB = 60°.
(ii) In the same diagram, draw a circle which passes through the points A, B and C and mark its centre O.
Answer
Steps to construct:
Step 1: Draw a line segment AB = 6 cm.
Step 2: At point A, draw a ray making an angle of 60o.
Step 3: With B as the center and radius 5.2 cm, draw an arc which intersects the ray at C.
Step 4: Join BC.
Step 5: Draw the perpendicular bisector of sides AB and BC which intersects each other at point O.
Step 6: With center as O and radius OA, draw a circle which touches through the points A, B, C.
5. Using ruler and compasses only, draw an equilateral triangle of side 5 cm and draw its inscribed circle. Measure the radius of the circle.
Answer
Steps to construct:
Step 1: Draw a line segment BC = 5 cm.
Step 2: With Center as B and radius 5 cm, with center as C and radius 5cm draw two arcs which intersect each other at point A.
Step 3: Join AB and AC.
Step 4: Draw the angle bisector of angles B and C which intersects each other at point I.
Step 5: From I, draw a perpendicular ID on BC.
Step 6: With center as I and radius ID, draw a circle which touches the sides of the triangle internally.
Step 7: On measuring the radius ID = 1.5 cm (approx).
6. (i) Conduct a triangle ABC with BC = 6.4 cm, CA = 5.8 cm and ∠ ABC = 60°. Draw its incircle. Measure and record the radius of the incircle.
(ii) Construct a ∆ABC with BC = 6.5 cm, AB = 5.5 cm, AC = 5 cm. Construct the incircle of the triangle. Measure and record the radius of the incircle.
Answer
Steps to construct:
Step 1: Draw a line segment BC = 6.4 cm.
Step 2: Construct an angle of 60o at point B.
Step 3: With C as center and radius CA = 5.8 cm, draw an arc cutting BD at A.
Step 4: Join AC.
Step 5: Draw the angle bisector of angle B and angle C which intersect each other at point O.
Step 6: Draw OE perpendicular to BC, intersecting BC at point E.
Step 7: With O as the center and OE as the radius draw the required incircle.
Step 8: On measuring the radius OE = 1.5 cm.
Steps to construct:
Step 1: Draw a line segment BC = 6.5 cm.
Step 2: With B as center and C as center draw arcs AB = 5.5 cm and AC = 5 cm.
Step 3: Join AB and AC.
Step 4: Draw the angle bisectors of B and C. The bisectors meet at point O.
Step 5: With O as the center. Draw an incircle which touches all the sides of the triangle ABC.
Step 6: From point O draw a perpendicular to side BC which cuts at point N.
Step 7: On measuring the radius ON = 1.5 cm.
7. The bisectors of angles A and B of a scalene triangle ABC meet at O.
(i) What is the point O called?
(ii) OR and OQ is drawn a perpendicular to AB and CA respectively. What is the relation between OR and OQ?
(iii) What is the relation between ∠ACO and ∠BCO?
Answer
(i) The point O where the angle bisectors meet is called the incenter of the triangle.
(ii) The perpendicular drawn from point O to AB and CA are equal. i.e., OR and OQ.
(iii) ∠ACO = ∠BCO. OC will bisect the ∠C.
8. Using ruler and compasses only, construct a triangle ABC in which BC = 4 cm, ∠ACB = 45° and the perpendicular from A on BC is 2.5 cm. Draw the circumcircle of triangle ABC and measure its radius.
Answer
Steps to construct:
Step 1: Draw a line segment BC = 4cm.
Step 2: At point B, draw a perpendicular and cut off BE = 2.5 cm.
Step 3: From, E, draw a line EF parallel to BC.
Step 4: From point C, draw a ray making an angle 45o which intersects EF at point A.
Step 5: Join AB.
Step 6: Draw the perpendicular bisectors of sides BC and AC intersecting each other at point O.
Step 7: Join OB, OC and OA.
Step 8: With O as the center and radius OB or OC or OA draw a circle which passes through points A, B, C.
Step 9: On measuring the radius OB = 2 cm.
9. Construct a regular hexagon of side 4 cm. Construct a circle circumscribing the hexagon.
Answer
Steps to construct:
Step 1: Draw a line segment AB = 4 cm.
Step 2: At points A and B draw rays making on angle of 120o each and cut off AF = BC = 4 cm.
Step 3: At point C and F draw rays making on angle of 120o each and cut off FE = CD = 4 cm.
Step 4: Join ED. The required ABCDEF hexagon is formed.
Step 5: Draw perpendicular bisectors of sides AB and BC intersecting each other at point O.
Step 6: With O as the center and radius equal OA or OB draw a circle which passes through the vertices of the hexagon.
10. Draw a regular hexagon of side 4 cm and construct its incircle.
Answer
Steps to construct:
Step 1: Draw a regular hexagon of sides 4 cm.
Step 2: Draw the angle bisector of A and B. which intersects each other at point O.
Step 3: Draw OL perpendicular to AB.
Step 4: With O as the center and radius OB, draw a circle which touches the sides of the hexagon.
Chapter Test
1. Draw a circle of radius 3 cm. Mark its centre as C and mark a point P such that CP = 7 cm. Using ruler and compasses only, Construct two tangents from P to the circle.
Answer
Steps of Construction:
Step 1: Draw a circle with centre C and radius 3 cm.
Step 2: Mark a point P such that CP = 7 cm.
Step 3: With CP as diameter, draw a circle intersecting the given circle at T and S.
Step 4: Joint PT and PS.
Step 5: Draw a tangent at Q to the circle given. Which intersects PT at D.
Step 6: Draw the angle bisector of ∠PDQ intersecting CP at E.
Step 7: With centre E and radius EQ, draw a circle. It will touch the tangent T and PS and the given circle at Q.
(i) a circle with AP as diameter.
(ii) two tangents to the above circle from the point Q.
Answer
Steps of construction:
Step 1: Draw a line segment AQ = 7 cm.
Step 2: From AQ, cut off AP = 4 cm
Step 3: With AP as diameter draw a circle with centre O.
Step 4: Draw bisector of OQ which intersect OQ at M.
Step 5: With centre M and draw a circle with radius MQ which intersects the first circle at T and S.
Step 6: Join QT and QS.
QT and QS are the tangents to the first circle.
3. Using ruler and compasses only, construct a triangle ABC having given c = 6 cm, b = 1 cm and ∠A = 30°. Measure side a. Draw carefully the circumcircle of the triangle.
Answer
Steps of Construction:
Step 1: Draw a line segment AC = 7 cm
Step 2: At C, draw a ray CX making an angle of 30°
Step 3: With centre A and radius6 cm draw an arc which intersects the ray CX at B.
Step 4: Join BA.
Step 5: Draw perpendicular bisectors of AB and AC intersecting each other at O.
Step 6: With centre O and radius OA or OB or OC, draw a circle which will pass through A, B and C.
This is the required circumcircle of ΔABC.
4. Using ruler and compasses only, construct an equilateral triangle of height 4 cm and draw its circumcircle.
Answer
Steps of Construction:
Step 1: Draw a line XY and take a point D on it.
Step 2: At D draw perpendicular and cut off DA = 4 cm.
Step 3: From A, draw rays making an angle of 30° on each side of AD meeting the line XY at B and C.
Now draw a circle perpendicular bisector of AC intersecting AD at O.
Step 4: Now draw perpendicular bisector of AC intersecting AD at O.
Step 5: With centre O and radius OA or OB or OC draw a circle which will pass through A, B and C. This is the required circumcircle of ΔABC.
5. Using ruler and compasses only:
(i) Construct a triangle ABC with the following data: BC = 7 cm, AB = 5 cm and ∠ABC = 45°.
(ii) Draw the inscribed circle to ΔABC drawn in part (i).
Answer
Steps of construction:
Step 1: Draw a line segment BC = 7 cm.
Step 2: At B, draw a ray BX making an angle of 45° and cut off BA = 5 cm.
Step 3: Join AC.
Step 4: Draw the angle bisectors of ∠B and ∠C intersecting each other at I.
Step 5: From I, draw a perpendicular ID on BC.
Step 6: With centre, I and radius ID, draw a circle which touches the sides of ΔABC at D, E and F respectively. This is the required inscribed circle.
6. Draw a triangle ABC, give that BC = 4 cm, ∠C = 75° and that radius of the circumcircle of ΔABC is 3 cm.
Answer
Steps of construction:
Step 1: Draw a line segment BC = 4 cm
Step 2: Draw the perpendicular bisector of BC.
Step 3: From B draw an arc of 3 cm radius which intersects the perpendicular bisector at O.
Step 4: Draw a ray CX making art angle of 75°.
Step 5: With centre O and radius 3 cm draw a circle which intersects the ray CX at A.
Step 6: Join AB. ΔABC is the required triangle.
7. Draw a regular hexagon of side 3.5 cm construct its circumcircle and measure its radius.
Answer
Steps of construction:
Step 1: Draw a regular hexagon ABCDF whose each side is 3.5 cm.
Step 2: Draw the perpendicular bisector of AB and BC which intersect each other at O.
Step 3: Join OA and OB.
Step 4: With centre O and radius OA or OB, draw a circle which passes through A, B, C, D, E and P. Then this is the required circumcircle.
8. Construct a triangle ABC with the following data: AB = 5 cm, BC = 6 cm and ∠ABC = 90°.
(i) Find a point P which is equidistant from B and C and is 5 cm from A. How many such points are there?
(ii) Construct a circle touching the sides AB and BC, and whose centre is equidistant from B and C.
Answer:
Steps of construction:
Step 1: Draw a line segment BC = 6 cm.
Step 2: At B, draw a ray BX making an angle of 90° and cut off BA = 5 cm.
Step 3: Join AC.
Step 4: Draw the perpendicular bisector of BC.
Step 5: From A with 5 cm radius draw arc which intersects the perpendicular bisector of BC at P and P’.
There are two points.
Step 6: Draw the angle bisectors of ∠B and ∠C intersecting at O
Step 7: From O, draw OD ⊥ BC.
Step 8: With centre O and radius OD, draw a circle which will touch the sides AB and BC. This is the required circle.
The solutions provided for Chapter 16 Construction of ML Aggarwal Textbook. This solutions of ML Aggarwal Textbook of Chapter 16 Construction contains answers to all the exercises given in the chapter. These solutions are very important if you are a student of ICSE boards studying in Class 10. While preparing the solutions, we kept this in our mind that these should based on the latest syllabus given by ICSE Board.
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