**ML Aggarwal ICSE Solutions Class 10 Math 16th Chapter Constructions**

Class 10 Chapter 16 Construction**Chapter 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.