Question

Ex. 4) The measures of the angles of the
triangle are in A. P. The smallest angle
is 40. Find the angles of the triangle in
degree and in radians.
triangle​

Answer 1

Answer:

Step-by-step explanation:

In an AP , the central number is the arithmatic mean of the number which succeeds it ( the cental number ) and precedes it . Here , let “ a “ be the least angle of the triangle , “ c “ be the largest angle , and “ b “ be the angle which lies between “ a “ and “ c “. AS THE ANGLES OF THE TRIANGLE ARE IN AP , b = ( a + c ) / 2

But a + b + c = 180 degrees ( angle sum property of a triangle )

Or a + ( a + c ) / 2 + c = 180

Or 3a + 3c = 360

Or a + c = 120 but c = 2a ( the least angle “ a “ being half of greatest angle “ c “ , as per condition of the problem ).

As a + c = 120 , therefore a + 2a = 120 or 3a = 120 or a = 120 / 3 = 40 , further c = 2a or 2 × 40 = 80 and b = ( a + c ) / 2 or b = ( 40 + 80 ) / 2 = 60 . Hence the required answer is 40 degrees , 60 degrees and 80 degrees .

We can verify our answer :-

40 , 60 and 80 are in AP , with commoon difference 20 . Here also , 80 = 2 × 40 ie the greatest angle 80 degree is double of the least angle 40 degree . Hence both the conditions posed in the problem are satisfied and above all 40 + 60 + 80 = 180 , the universal angle sum property of a triangle is satisfied too .