# 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:

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

Proof:

In ΔBB_{4}C', B_{3}C || B_{4}C'

Hence by Basic proportionality theorem,

B_{3}B_{4}/BB_{3} = 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

**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:

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

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