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A triangle if its perimeter is 10.4 cm and two angles are 45° and 120°. Construct the following and give justification

Solution:

Consider ABC as a triangle of perimeter 10.4 cm

AB + BC + CA = 10.4 cm and two angles are 45° and 120°

Let us take ∠B = 45° and ∠C = 120°

Steps of Construction

1. Construct a line segment XY of the given length (perimeter of the required triangle) 10.4 cm

2. Construct angle ∠LXY = ∠B = 45° and ∠MYX = ∠C = 120°

3. Now bisect ∠LXY and ∠MYX and let these bisectors intersect at A

4. Construct perpendicular bisectors PQ and RS of AX and AY

5. Let PQ line segment XY at the point B and RS intersect line segment XY at the point C

6. Join AB and AC

7. Triangle ABC is the required triangle

A triangle if its perimeter is 10.4 cm and two angles are 45° and 120°. Construct the following and give justification

Justification

We know that

B lies on perpendicular bisector PQ of AX

AB + BC + CA = XB + BC + CY = XY

In triangle AXB, AB = XB

So ∠BAX = ∠AXB …. (1)

As ∠ABC is exterior angle of triangle AXB

∠ABC = ∠BAX + ∠AXB

Using equation (1)

∠ABC = ∠AXB + ∠AXB

∠ABC = 2 ∠AXB

∠ABC = ∠LXY (AX is the bisector of ∠LXB)

In triangle AYC, AC = CY

∠CAY = ∠AYC

As ∠ACB is an exterior angle of triangle AYC

∠ACB = ∠CAY + ∠AYC

∠ACB = ∠CAY + ∠CAY

∠ACB = 2∠CAY

∠ACB = ∠MYX (AY is the bisector of ∠MYX)

Therefore, the construction is justified.

✦ Try This: Construct each of the following and give justification: A triangle if its perimeter is 12.4 cm and two angles are 60° and 110°.

☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 11

NCERT Exemplar Class 9 Maths Exercise 11.4 Problem 1

Summary:

A triangle if its perimeter is 10.4 cm and two angles are 45° and 120° is constructed and justified

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