Class 10 RD Sharma Solutions - Chapter 11 Constructions - Exercise 11.2 | Set 2

In Chapter 11 of Class 10 RD Sharma, we delve into the topic of "Constructions" where students learn the practical aspects of geometry by drawing various geometric figures. This chapter helps students understand the step-by-step methods to construct specific angles, triangles, and other shapes using geometric tools like a compass, ruler, and protractor. Mastery of these techniques is essential for developing precision and accuracy in geometric constructions.

Constructions

The Constructions involve creating precise geometric figures using only a compass and a straightedge. This chapter focuses on constructing angles, bisectors, and triangles based on the given measurements. By following systematic steps students can accurately draw shapes that meet the specific requirements outlined in the exercises.

Question 11. Construct a triangle similar to a given ΔXYZ with its sides equal to (3/2)^th of the corresponding sides of ΔXYZ. Write the steps of construction.                     

Solution:

Follow these steps for the construction:

Step 1: Construct a triangle XYZ along some feasible data.

Step 2: Construct a ray YL creating an acute angle along XZ and break off 5 equal parts creating 

YY₁= Y₁Y₂ = Y₂Y₃ = Y₃Y₄.

Step 3: Connect Y₄ and Z.

Step 4: From Y₃, construct Y₃Z’ parallel to Y₄Z and Z’X’ parallel to ZX.

Therefore,

We have the required triangle ΔX’YZ’.

Question 12. Draw a right triangle in which sides (other than the hypotenuse) are of lengths 8 cm and 6 cm. Then construct another triangle whose sides are 3/4 times the corresponding sides of the first triangle.

Solution:

Follow these steps for the construction

Step 1: Construct right ΔABC right angle at B and BC of 8 cm and BA of 6 cm.

Step 2: Construct a line BY creating a cut angle along BC and break off 4 equal parts.

Step 3: Connect 4C and Construct 3C’ || 4C and C’A’ parallel to CA.

Therefore,

We have the required triangle ΔBC’A’ is the required triangle.

Question 13. Construct a triangle with sides 5 cm, 5.5 cm, and 6.5 cm. Now construct another triangle, whose sides are 3/5 times the corresponding sides of the given triangle.

Solution:

Follow these steps for the construction:

Step 1: Construct a line segment BC of 5.5 cm.

Step 2: Along centre B and radius 5 cm and along centre C and radius 6.5 cm, 

Construct arcs that bisect each other at A

Step 3: Connect BA and CA. ΔABC is the given triangle.

Step 4: At B, construct a ray BX creating an acute angle and break off 5 equal parts from BX.

Step 5: Connect C5 and Construct 3D || 5C which connects BC at D.

From D, construct DE || CA which meets AB at E.

Therefore,

We have the required triangle ΔEBD.

Question 14. Construct a triangle PQR with side QR = 7 cm, PQ = 6 cm and ∠PQR = 60°. Then construct another triangle whose sides are 3/5 of the corresponding sides of ΔPQR. 

Solution:

Follow these steps for the construction:

Step 1: Construct a line segment QR = 7 cm.

Step 2: At Q construct a ray QX creating an angle of 60° and cut of PQ = 6 cm. 

Connect PR.

Step 3: Construct a ray QY creating an acute angle and break off 5 equal parts.

Step 4: Connect 5, R and through 3, construct 3, S parallel to 5, R which meet QR at S.

Step 5: Through S, construct ST || RP meeting PQ at T.

Therefore,

We have the required ΔQST.

Question 15. Construct a ΔABC in which base BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then construct another triangle whose sides are 3/4 of the corresponding sides of ΔABC.  

Solution:

Follow these steps for the construction:

Step 1: To construct a triangle ABC 

Along side of BC of 6 cm, 

AB of 5 cm and ∠ABC = 60°.

Step 2: Construct a ray BX, which creates an acute angle ∠CBX below the line BC.

Step 4: Make four points B₁, B₂, B₃ and B₄ on BX such that BB₁ = B₁B₂  = B₂B₃ = B₃B₄.

Step 5: Connect B₄C and construct a line through B₃ parallel to B₄C bisecting BC to C’.

Step 6: Construct a line through C’ parallel to the line CA to intersect BA at A’.

Therefore,

We have the required triangle ΔABC.

Question 16. Construct a right triangle in which the sides (other than the hypotenuse) arc of lengths 4 cm and 3 cm. Now, construct another triangle whose sides are 5/3 times the corresponding sides of the given triangle.

Solution:

Follow these steps for the construction:

Step 1: Construct a right triangle ABC in which the sides (other than the hypotenuse) are 

of lengths 4 cm and 3 cm. ∠B = 90°.

Step 2: Construct a line BX, that creates an acute angle ∠CBX below the line BC.

Step 3: Mark 5 points B₁, B2₂, B₃, B₄ and B₅ on BX such that BB₁ = B₁B₂ = B₂B₃ = B₃B₄ = B₄B₅.

Step 4: Connect B₃ to C and Construct a line through B₅ parallel to B₃C, 

bisecting the extended line segment BC at C’.

Step 5: Construct a line through C’ parallel to CA bisecting the extended line segment BA at A’.

Therefore,

We have the required triangle ΔA'BC'.

Question 17. Construct a ΔABC in which AB = 5 cm, ∠B = 60°, altitude CD = 3 cm. Construct a ΔAQR similar to ΔABC such that side of ΔAQR is 1.5 times that of the corresponding sides of ΔACB.

Solution:

Follow these steps for the construction:

Step 1: Construct a line segment AB of 5 cm.

Step 2: At A, construct a perpendicular and break off AE = 3 cm.

Step 3: From E, construct EF || AB.

Step 4: From B, construct a ray creating an angle of 60 meeting EF at C.

Step 5: Connect CA. After that ABC is the triangle.

Step 6: From A, construct a ray AX creating an acute angle along AB and break 

off 3 equal parts creating AA₁ = A₁A₂ = A₂A₃.

Step 7: Connect A₂ and B.

Step 8: From A , construct A^B’ parallel to A₂B and B’C’ parallel to BC.

Therefore,

We have the required triangle ΔC’AB’.

Read More:

• Construction of Triangles
• Constructions in LaTeX

Conclusion

In Exercise 11.2 | Set 2 of Chapter 11, "Constructions" from RD Sharma's Class 10 Mathematics students practice fundamental construction techniques essential for the accurate geometrical problem-solving. The exercise reinforces the understanding of the constructing triangles, bisectors and other geometric figures using the compass and ruler. Mastering these skills is crucial for the higher-level geometry and helps build a solid foundation for the future mathematical studies.