Draw a triangle ABC with side BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then construct a triangle whose sides are 3/4 of the corresponding sides of the triangle ABC.

Solution:

• Draw the triangle with the given conditions.

• Then draw another line that makes an acute angle with the baseline. Divide the line into m + n parts where m and n are the ratios given.

• Two triangles are said to be similar if their corresponding angles are equal, are said to satisfy Angle-Angle-Angle (AAA) Axiom.

• The basic proportionality theorem states that “If a straight line is drawn parallel to a side of a triangle, then it divides the other two sides proportionally".

Steps of constructions:

1. Draw a line BC = 6 cm.
2. At B, make ∠C = 60° and cut an arc at A on the same line so that BA = 5 cm. Join AC, ΔABC is obtained.
3. Draw the ray BX such that ∠CBX is acute.
4. Mark 4 (since 4 > in 3/4) points B₁, B₂, B₃, B₄ on BX such that BB₁ = B₁B₂ = B₂B₃ = B₃B₄
5. Join B₄ to C and draw B₃C' parallel to B₄C to intersect BC at C'.
6. Draw C'A' parallel to CA to intersect BA at A’.

Now, ΔA'BC' is the required triangle similar to ΔABC where BA'/BA = BC'/BC = C'A'/CA = 3/4

Proof:

In ΔBB₄C , B₃C' || B₄C

Hence by Basic proportionality theorem,

B₃B₄/BB₃ = C'C/BC' = 1/3

C'C /BC' + 1 = 1/3 + 1

(C'C + BC')/BC' = 4/3

BC/BC' = 4/3 or BC'/BC = 3/4

Consider ΔBA'C' and ΔBAC

∠A'BC' = ∠ABC = 60°

∠BCA' = ∠BCA (Corresponding angles ∵ C'A'||CA)

∠BA'C' = ∠BAC (Corresponding angles)

By AAA axiom, ΔBA'C' ~ ΔBAC

Therefore, corresponding sides are proportional,

BC'/BC = BA'/BA = C'A'/CA = 3/4

☛ Check: NCERT Solutions for Class 10 Maths Chapter 11

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Video Solution:

Draw a triangle ABC with side BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then construct a triangle whose sides are 3/4 of the corresponding sides of the triangle ABC

NCERT Solutions Class 10 Maths Chapter 11 Exercise 11.1 Question 5

Summary:

A triangle ABC of sides 6cm, 5 cm and ∠ABC = 60° and another triangle A'BC' of sides 3/4 of the corresponding sides of triangle ABC have been constructed. 

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☛ Related Questions:

• Draw a line segment of length 7.6 cm and divide it in the ratio 5:8. Measure the two parts.
• Construct a triangle of sides 4 cm, 5 cm and 6 cm and then a triangle similar to it whose sides are 2/3 of the corresponding sides of the first triangle.
• Construct a triangle with sides 5 cm, 6 cm and 7 cm and then another triangle whose sides are 7/5 of the corresponding sides of the first triangle.
• Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then another triangle whose sides are 1(1/2) times the corresponding sides of the isosceles triangle.