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 AngleAngleAngle (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:
 Draw a line BC = 6 cm.
 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.
 Draw the ray BX such that ∠CBX is acute.
 Mark 4 (since 4 > in 3/4) points B₁, B₂, B₃, B₄ on BX such that BB₁ = B₁B₂ = B₂B₃ = B₃B₄
 Join B₄ to C and draw B₃C' parallel to B₄C to intersect BC at C'.
 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
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.
☛ 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.
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