Question

The angles of a triangle are in AP. The greatest angle is twice the least. Find the difference between the largest and smallest angle.

Answer 1

Hint: Here, we can first take the three angles of the triangle as (a-d), a and (a+d). Then, we can find the values of a and d using the given conditions. Hence, we can find the values of the angles of the triangle and also the difference between the largest and smallest angle.

Complete step-by-step answer:

We know that any three numbers in AP are considered as:

a-d, a and a+d

Here, a is any term of the AP and d is a common difference of the AP.

We can easily say that the greatest among these three is (a+d) and the smallest one is (a-d).

Since, it is given that the greatest angle is twice the least angle. So, we can write:

 a+d=2$\times$( a-d ) 

$\Rightarrow a+d=2a-2d$

$\Rightarrow d+2d=2a-a$

$\Rightarrow 3d=a$

$\Rightarrow a=3d...........\left(1\right)$

Now, we know that according to the angle sum property of a triangle, the sum of all the three angles of a triangle is equal to 180 degrees.

Therefore, the sum of (a+d), a and (a-d) must be equal to 180. So, we can write:

 a+d+a+a-d=180 

$\Rightarrow 3a=180$

$\Rightarrow a={\displaystyle \frac{180}{3}}$

 $\Rightarrow a=60$

So, the value of a is 60. Therefore, the measure of one of the angles is 60 degrees.

Now, on putting a = 60 in equation (1), we get:

 60=3d 

$\Rightarrow d={\displaystyle \frac{60}{3}}=20$

So, the other two angles are:

(60-20) = 40 degrees and (60+20) = 80 degrees.

So, we get that the three angles of this triangle are 40, 60 and 80 degrees.

Difference between the largest and the smallest angle is = (80-40) = 40 degrees

Hence, the difference between the largest and the smallest angle is equal to 40 degrees.

Note: Students should remember that three numbers in an AP are always taken as (a-d), a and (a+d). Students should also remember the angle sum property of a triangle to avoid unnecessary mistakes.