Question

The sides of a triangle are given by x - 2y = 1, 2x - y = 6, 3x - y = 8 , then if the vertex having largest interior angle of this triangle is (α , ß) then α + ß is …………..

Answer 1

Given : The sides of a triangle are given ; x - 2y = 1 , 2x - y = 6 and 3x - y = 8. the vertex having largest interior angle of this triangle is (α, β)

To find : the value of α + β

solution : let ABC is the triangle.

where AB : x - 2y = 1 , BC : 2x - y = 6 and CA : 3x - y = 8

solving AB and BC gives Point B

2(x - 2y) - (2x - y) = 2 × 1 - 6

⇒2x - 4y - 2x + y = -4

⇒-3y = -4

⇒y = 4/3 and x = 1 + 2y = 1 + 8/3 = 11/3 ,i.e., B = (11/3, 4/3)

solving BC and CA gives point C

2x - y - (3x - y) = 6 - 8

⇒-x = - 2

⇒x = 2 and y = 2x - 6 = -2, i.e., C = (2, -2)

solving CA and AB gives points A

(x - 2y) - 2(3x - y) = 1 - 2 × 8

⇒x - 6x = - 15

⇒x = 3 and y = (x - 1)/2 = 1 , I.e., A = (3, 1)

now, length AB = √{(3 - 11/3)² + (1 - 4/3)²}

= √{4/9 + 1/9} = √5/3

length BC = √{(11/3 - 2)² + (4/3 + 2)²}

= √{25/9 + 100/9} = 5√5/3

length CA = √{(3 - 2)² + (1 + 2)²} = √10

it is clear that CA is the largest side of ABC

so, B is largest interior angle.

therefore (α, β) = (11/3, 4/3)

now, the value of α + β = 11/3 + 4/3 = 5