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​

Answer 1

Answer:

• 90°, 60° & 30° is the required angles of triangle.

Step-by-step explanation:

According to the Question

As Per given Condition :-

• 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

Let the largest angle be x

Smallest angle be x/3

And another angle be y

1st equation :-

$:\implies$ x = x/3 + y

$:\implies$ x = x +3y/3

$:\implies$ 3x = x + 3y

$:\implies$ 2x = 3y

$:\implies$ 2x - 3y = 0 ⠀⠀⠀⠀⠀⠀⠀⠀----(i)

Now,

2nd equation :-

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

$:\implies$ x + x/3 +y = 180°

$:\implies$ 3x + x + 3y = 540°

$:\implies$ 4x + 3y = 540° ⠀⠀⠀⠀⠀⠀⠀⠀---(ii)

Adding equation (i) & (ii) we get

$:\implies$ 6x = 540°

$:\implies$ x = 540/6

$:\implies$ x = 90°

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

$:\implies$ 180° -3y = 0

$:\implies$ -3y = -180°

$:\implies$ y = 180°/3

$:\implies$ y = 60°

And , smallest angle = x/3 = 90°/3 = 30°

• Hence, the angles of triangles are 90° , 60° & 30°.

Answer 2

Answer:

Given :-

• The largest angle of a triangle is equal to the sum of the other two angles.
• The smallest angle is ⅓ of the largest angle.

To Find :-

• What is the angles of a triangle.

Solution :-

Let,

➲ Largest Angle = a

➲ Smallest Angle = a/3

➲ Other Angle = b

✭ In the first case :-

$\leadsto \sf a =\: \dfrac{a}{3} + b$

$\leadsto \sf a =\: \dfrac{a + 3b}{3}$

By doing cross multiplication we get,

$\leadsto \sf 3a =\: a + 3b$

$\leadsto \sf 3a - a =\: 3b$

$\leadsto \sf 2a =\: 3b$

$\leadsto \sf\bold{\purple{2a - 3b =\: 0\: ------\: (Equation\: No\: 1)}}$

✭ In second case :

Now, as we know that :

$\footnotesize\clubsuit\: \: \sf\boxed{\bold{\pink{Sum\: Of\: All\: Angles_{(Triangle)} =\: 180^{\circ}}}}\: \: \bigstar$

According to the question by using the formula we get,

$\leadsto \sf a + \dfrac{a}{3} + b =\: 180^{\circ}$

$\leadsto \sf \dfrac{3a + a + 3b}{3} =\: 180^{\circ}$

$\leadsto \sf \dfrac{4a + 3b}{3} =\: 180^{\circ}$

By doing cross multiplication we get,

$\leadsto \sf 4a + 3b =\: 3(180^{\circ})$

$\leadsto \sf\bold{\purple{4a + 3b =\: 540^{\circ}\: ------\: (Equation\: No\: 2)}}$

Now, by adding both equation we get,

$\leadsto \sf 2a - 3b + 4a + 3b =\: 0 + 540^{\circ}$

$\leadsto \sf 2a + 4a {\cancel{- 3b}} {\cancel{+ 3b}} =\: 540^{\circ}$

$\leadsto \sf 6a =\: 540^{\circ}$

$\leadsto \sf a =\: \dfrac{\cancel{540^{\circ}}}{\cancel{6}}$

$\leadsto \sf\bold{\green{a =\: 90^{\circ}}}$

Again, by putting the value of a in the equation no 2 we get,

$\leadsto \sf 4a + 3b =\: 540^{\circ}$

$\leadsto \sf 4(90^{\circ}) + 3b =\: 540^{\circ}$

$\leadsto \sf 360^{\circ} + 3b =\: 540^{\circ}$

$\leadsto \sf 3b =\: 540^{\circ} - 360^{\circ}$

$\leadsto \sf 3b =\: 180^{\circ}$

$\leadsto \sf b =\: \dfrac{\cancel{180^{\circ}}}{\cancel{3}}$

$\leadsto \sf\bold{\green{b =\: 60^{\circ}}}$

Hence, the required angles of a triangle are :

❒ Largest Angle Of Triangle :

$\implies \sf Largest\: Angle_{(Triangle)} =\: a$

$\implies \sf\bold{\red{Largest\: Angle_{(Triangle)} =\: 90^{\circ}}}$

❒ Smallest Angle Of Triangle :

$\implies \sf Smallest\: Angle_{(Triangle)} =\: \dfrac{a}{3}$

$\implies \sf Smallest\: Angle_{(Triangle)} =\: \dfrac{\cancel{90^{\circ}}}{\cancel{3}}$

$\implies \sf\bold{\red{Smallest\: Angle_{(Triangle)} =\: 30^{\circ}}}$

❒ Other Angle Of Triangle :

$\implies \sf Other\: Angle_{(Triangle)} =\: b$

$\implies \sf\bold{\red{Other\: Angle_{(Triangle)} =\: 60^{\circ}}}$

$\therefore$ The angles of a triangle is 90°, 30°, 60° respectively.