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

Solution:

• Draw the line segment of the largest length 7 cm. Measure 5 cm and 6 cm separately and cut arcs from both the ends of the line segment such that they cross each other at one point. Connect this point from both the ends

• Then draw another line that makes an acute angle with the given line (7 cm). 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 and are said to satisfy the AA criteria.

• 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 with B and C as centres and radii 5 cm and 6 cm respectively. Draw arcs to intersect at A. ΔABC is obtained.
2. Draw ray BX making ∠CBX an acute angle.
3. Mark 7 points (greater of 7 and 5 in 7/5 ) B₁, B₂,………B₇ on BX such that BB₁ = B₁B₂ =............... = B₆B₇
4. Join B₅ (smaller of 7 and 5 in 7/5 which is the 5^th point) to C and draw B₇C' parallel to B₅C intersecting the extension of BC at C'.
5. Through C' draw C'A' parallel to CA to meet the extension of BA at A’. Now, ΔA' B'C' is the required triangle similar to ΔABC where BA'/BA = C'A'/CA = BC'/BC = 7/5

Proof:

In ΔBB₇C', B₃C is parallel to B₇C'

Hence by Basic proportionality theorem,

B₅B₇/BB₅ = CC'/BC = 2/5

Adding 1 to both the sides of CC'/BC = 2/5,

CC'/BC + 1 = 2/5 + 1

(BC + CC')/BC = 7/5

BC'/BC = 7/5

Consider ΔBAC and ΔBA'C'

∠ABC = ∠A'BC' (Common)

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

By AA criteria, ΔBAC ∼ ΔBA'C'

∴ Corresponding sides are proportional

Hence,

BA'/BA = C'A'/A = BC'/BC = 7/5

☛ Check: NCERT Solutions for Class 10 Maths Chapter 11

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Video Solution:

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.

NCERT Solutions Class 10 Maths Chapter 11 Exercise 11.1 Question 3

Summary:

A triangle BAC of sides 5 cm, 6 cm, and 7 cm and another triangle B'A'C whose sides are 7/5 of the corresponding sides of the first triangle BAC have been constructed

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