ML Aggarwal ICSE Solutions Class 10 Math 16th Chapter Constructions
Class 10 Chapter 16 ConstructionChapter 16 Construction
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 so that op = 7 cm
Construct the tangents to the circle from P. Measure and record the length of one of the tangents.
Solution:
- Initially draw a circle with centre ‘O’ and radius 4 cm.
- Take one point P from the centre of a circle at a distance of 7 cm i.e. op = 7 cm.
- Now, bisect op at point ‘M’ and with centre M & diameter op draw another circle which intersects given circle at points A and B.
- Now, joint the points PA and PB which are the pairs of tangents to the circle.
- If we measure PA it is found that l (PA) = 5. 5 cm
(2) Draw a line AB = 6m. 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 P circle with AB as a diameter. Measure the length of each tangent.
Solution:
We follow the following steps for construction:
- First draw a line segment AB = 6 cm as shown.
- Draw a perpendicular bisector of AB which bisects AB at ‘o’.
- Now, take point ‘o’ as centre and radius OB to draw a circle.
- Now extend AB to point ‘P’ so that OP = 5 cm.
- Draw Perpendicular bisector of op which meets act M.
- With centre M and by taking radius as OM draw a circle. Which intersects the given circle at points T and S as shown.
- Now, join OT, OS, TP, and, SP. We can see here, PT and PS are the required tangents to the circle.
- Here, it is found that l (PT) = l (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.
Solution:
We follow the following steps for construction:
- Take a point ‘O’ as centre and draw two concentric. Circles with radii 4 cm and 6 cm as shown.
- Now join OA and take it mid – point as M.
- By taking point M as centre and MA as a radius draw a another circle which intersects given circle at points P and Q as shown.
- Now, join Segments AP and AQ. Here, AP and AQ are the required tangent to the given circle from point A.
(4) Draw a circle of radius 3cm. Take two points P and Q on one of its extended diameter each at a distance of 7 cm from its centre. Draw tangents to the circle from these two Points P and Q.
Solution:
We follow the following steps for construction:
- Initially take a point ‘O’ as centre and draw a circle of radius 3 cm as shown.
- extend the diameter from both sides so that OP = OQ = 7 cm.
- Let us consider mid – points of OP and OQ as M and N respectively as shown.
- Now take Points M and N as centre and OP & OQ. As diameters to draw circle which intersects given. circle at points A, B and C and D respectively.
- Now join the segments PA, PB, QC and QD as shown. Here, PA, PB and QC, QD are the required tangents.
Exercise 16.2
(1) Draw an equilateral at triangle of side 4 cm. Draw its circumcircle.
Solution:
We follow the following steps for construction:
- Initially, draw a line segment BC = 4 cm as shown.
- Now by taking points B and C as centres draw two areas of radius 4 cm which intersects each other at point A as shown.
- Now, join segments AB and AC & hence △ABC is formed which is a equilateral triangle.
- Now, draw right bisectors of BC and AC intersecting each other at point O and join segments OA, OB & OC.
- By taking centre ‘O’ and radius OB or OC or OA draw a circle which passes through points A, B and C.
Now, here we got the required circumcircle of given △ABC.
(2) Using a ruler & a pair of compass only, construct.
(i) a triangle ABC given AB = 4 cm, BC = 6 cm, ∠ABC = 90°
(ii) a circle which passes through points A, B and C and marks its centre as ‘O’.
Solution:
We follow the following steps for construction:
- Initially, draw a line segment AB = 4 cm as shown.
- At Point B, draw a ray BX making an angle of 90° so that BC = 6 cm and join AC.
- Now, draw Perpendicular bisectors of sides AB & AC. Which intersects each other at point ‘O’ as shown.
- By taking ‘O’ as centre and radius equal to OB or OA or OC draw a circle which passes through Points A, B and C respectively.
- Here, we got our required circle.
(3) Construct a triangle with sides 3cm, 4cm, and 5cm. Draw its circumcircle and measure its radius.
Solution:
We follow following steps for construction.
- Initially draw a line segment BC = 4 cm as shown.
- By taking point B as centre and radius 3 cm, also by taking point C as centre with radius 5 cm draw two areas which intersects each other at one point say ‘A’.
- Now, join segments AB and AC.
- Now, draw perpendicular bisector of sides BC & AC. Which intersects each other at point ‘O’ and join OB.
- By taking point ‘O’ as centre and OB as radius draw a circle which passes through point A, B and C as shown.
- After measuring the radius its found that OB = 2.5 cm.
(4) Using ruler & compass only:
(i) Construct a triangle ABC with the following date:
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’.
Solution:
We follow following steps for construction:
- Initially, draw a line segment AB = 6 cm as shown.
- at point A, draw a ray making angle 60°
- By taking point B as centre and radius equal to 5.2 cm draw an are which intersects the ray at point C as shown.
- Now, join BC and draw perpendicular bisectors of AB and BC which intersects each other at point ‘O’.
- By taking point ‘O’ as centre and OA as radius draw a circle which touches the △ABC at points A, B & C as shown in fig below.
(5) Using ruler & compass only, draw an equilateral triangle of side 5 cm and draw its inscribed circle. Measure the radius of the circle.
Solution:
We follow the following steps for construction:
- Initially, draw a line segment BC = 5 cm as shown.
- By taking points B and C as centers and radius equal to 5 cm draw two areas which intersects each other at pt A.
- Now, join segments AB and AC as shown.
- Now, draw angle bisectors of ∠B and ∠C which intersects each other at point I.
- Now, from point I, draw perpendicular ID on BC.
- Now, by taking I as centre & radius ID draw a circle which touches the sides of triangle internally.
- Here, it is the required in circle formed. and ID = 1.5 cm.
(7) Using ruler & 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 circum circle of triangle ABC & measure its radius.
Solution:
We follow the following steps for construction.
- Initially draw a line segment BC = 4 cm.
- At point B, draw a perpendicular so that BE = 2.5 cm
- From point E, draw a line EF || BC.
- From point C, draw a ray making angle of 45° which intersects EF at point A as shown.
- Now join AB & draw perpendicular bisectors of sides BC & AC which intersects each other at point O as shown.
- Join OB, OC and OA.
- By taking ‘O’ as centre and radii equal to OB or OC or OA draw a circle which touches A at points A, B & C.
- Now, Here the circum circle of △ABC is formed and OB = 2 cm.
(8) Using the ruler & Compass only.
Construct a △ABC so that BC = 5 cm & AB = 6.5 cm and ∠ACB = 120° (i) Construct a circumcircle of △ABC
(ii) Construct a cyclic quadrilateral ABCD, so that point D is equidistant from AB and BC.
Solution:
We follow the following steps for construction:
- Initially, draw a line segment AB = 6.5 cm
- At point B, construct an angle of 120° so that BC = 5 cm. Now, join segment AC and △ABC is formed.
- Now, draw perpendicular bisectors of sides AB & BC. which intersects each other at point ‘O’.
- By taking point ‘O’ as centre & radius equal to OB or OC or OA draw a circumcircle of △ABC.
- Extend perpendicular of bisector of AB & it Mtersects.
- Now, join AD and CD. Thus, quadrilateral ABC is formed which is cyclic.
(10) Draw a regular hexagon of side 4 cm and construct its incircle.
Solution:
We follow the following steps for construction:
- Initially draw a regular hexagon ABCDEF of side 4 cm.
- Draw the angle bisectors of ∠A and ∠B which intersects each other at point ‘O’.
- Draw OL ⊥or AB
- By taking O as centre and radius equal to OB draw a circle which touches the sides of hexagon.
