Draw a triangle ABC with side BC = 7 cm, ∠B = 45°, ∠A = 105°. Then, construct a triangle whose sides are 4/3 times the corresponding sides of ΔABC

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. They 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 construction:

1. Draw BC = 7cm and at B, make an angle ∠CBY = 45° and at C, make ∠BCZ = 30° [Since, 180° - (45° + 105° )]. Both BY and CZ intersect at A and thus ΔABC is constructed.
2. Draw the ray BX so that ∠CBX is acute.
3. Mark 4 (since 4 > 3 in 4/3) points B₁, B₂, B₃, B₄ on BX such that BB₁ = B₁B₂ = B₂B₃ = B₃B₄
4. Join B₃ third point on BX, (since 3 < 4 in 4/3) to C and draw B C' parallel to BC such that C’ lies on the extension of BC.
5. Draw C’A’ parallel to CA to intersect the extension of BA at A’. Now, triangle A'BC' is the required triangle similar to ΔABC where, BA'/BA = BC'/BC = C'A'/CA = 4/3

Proof:

In ΔBB₄C', B₃C || B₄C'

Hence by Basic proportionality theorem,

B₃B₄/BB₃ = CC'/BC = 1/3

CC'/BC + 1 = 1/3 + 1 (Adding 1)

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

BC'/BC = 4/3

Consider ΔBA'C' and ΔBAC

∠A'BC' = ∠ABC = 45°

∠BC'A' = ∠BCA = 30° (Corresponding angles as CA|| C'A')

∠BA'C' = ∠BAC = 105° (Corresponding angles)

By AAA axiom, ΔBA'C' ~ ΔBAC

Hence corresponding sides are proportional

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

☛ Check: NCERT Solutions for Class 10 Maths Chapter 11

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Video Solution:

Draw a triangle ABC with side BC = 7 cm, ∠B = 45°, ∠A = 105°. Then, construct a triangle whose sides are 4/3 times the corresponding sides of ΔABC.

NCERT Solutions Class 10 Maths Chapter 11 Exercise 11.1 Question 6

Summary:

A triangle ABC with a side 7 cm,∠B = 45°,∠A = 105°, and another triangle A'BC' of sides that is 4/3 times the corresponding sides of triangle ABC have been constructed.

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☛ Related Questions:

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