Question

The largest angle of a triangle is equal to the sum of the other two angles. If the smallest angle is 1/3 of the largest angle then the angles of a triangle is


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Answer 1

Given:

The largest angle of a triangle is equal to the sum of the other two angles. The smallest angle is 1/3 of the largest angle.

Need to Find: Angles of Triangle.

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❍ Now, let us assume that tLargest angle as x, Smallest angle as x/3 and another angle as y

1st equation :-

$\\:\implies\quad\sf{x = \dfrac{x}{3} + y}$

$\\:\implies\quad\sf{x = x +\dfrac{3y}{3}}$

$\\:\implies\quad\sf{3x = x + 3y}$

$\\:\implies\quad\sf{2x = 3y}$

$\\:\implies\quad\sf{2x - 3y = 0\qquad\qquad\left\lgroup\begin{matrix}\sf{{eq}^{n}} \: (I) \end{matrix}\right\rgroup\]}$

Now,

2nd equation :-

As we know that sum of all angles in a triangle is 180°.

$\\:\implies\quad\sf{x + x/3 +y = {180}^{o}}$

$\\:\implies\quad\sf{3x + x + 3y = {540}^{o}}$

$\\:\implies\quad\sf{4x + 3y = {540}^{o}\qquad\quad\left\lgroup\begin{matrix}\sf{{eq}^{n}} \: (II) \end{matrix}\right\rgroup\]}$

Adding equation (i) & (ii) we get

$\\:\implies\quad\sf{6x = {540}^{o}}$

$\\:\implies\quad\sf{x =\dfrac{540}{6}}$

$\\:\implies\quad\underline{\boxed{\pmb{\frak{x = {90}^{o}}}}}$

Now, putting the value of x = 90° in Equation (i) we get

$\\:\implies\quad\sf{{180}^{o} -3y = 0}$

$\\:\implies\quad\sf{-3y = {-180}^{o}}$

$\\:\implies\quad\sf{y =\dfrac{{180}^{o}}{3}}$

$\\:\implies\quad\underline{\boxed{\pmb{\frak{y = {60}^{o}}}}}$

And ,

$\\:\implies\quad\sf{Smallest\: angle =\dfrac{x}{3}}$

$\\:\implies\quad\sf{Smallest\: angle =\dfrac{90}{3}}$

$\\:\implies\quad\underline{\boxed{\pmb{\frak{Smallest\: angle={30}^{o}}}}}$

⠀

$\qquad\leadsto$ Hence, the angles of triangles are 90° , 60° & 30°.

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