Question

If one angle is the average of the other two angles and the difference between the greatest and least angles is 60 degree, then the angles of the triangles are in the ratio_____ .

Answer 1

Answer:3:1:2

Step-by-step explanation:

Answer 2

Answer:  The angles of the triangles are in the ratio 3 : 2 : 1.

Step-by-step explanation:  Given that one angle of a triangle is the average of the other two angles and the difference between the greatest and least angles is 60 degree.

We are to find the ratio of the measures of the three angles of the triangle.

Let x,  y  and  z represents the measures in degrees of the three angles of the given triangle.

Then, according to the given information, we have

$y=\dfrac{x+z}{2}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)$

and

$x-z=60^\circ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)$

Also, we know that the sum of the measures of three angles of a triangle is 180°, so we have

$x+y+z=180^\circ\\\\\\\Rightarrow x+\dfrac{x+z}{2}+z=180^\circ~~~~~~~~~~~~~~~~~~~~~~~~~~[\textup{Using equation (i)}]\\\\\\\Rightarrow 3x+3z=2\times180^\circ\\\\\Rightarrow x+z=120^\circ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(iii)$

Adding equations (ii) and (iii), we get

$(x+z)+(x-z)=60^\circ+120^\circ\\\\\Rightarrow 2x=180^\circ\\\\\Rightarrow x=\dfrac{180^\circ}{2}\\\\\Rightarrow x=9<strong>0^\circ.$

From equation (ii), we get

$90^\circ-z=60^\circ\\\\\Rightarrow z=90^\circ-60^\circ\\\\\Rightarrow z=30^\circ.$

From equation (i), we get

$y=\dfrac{90^\circ+30^\circ}{2}=\dfrac{120^\circ}{2}=60^\circ.$

Therefore, the ratio of the measures of the three angles is given by

$x:y:z=90^\circ:60^\circ:30^\circ=3:2:1.$

Thus, the angles of the triangles are in the ratio 3 : 2 : 1.