Question

If three angles in a triangle ABC are in the ratio 1:2:1. Find all the angles of the triangle. Classify the triangle in two different ways.

Answer 1

right angled isosceles triangle

Solution

Angles are in ratio 1:2:1

sum of interior angle of triangle = 180

$x + 2x + x = 180$

$x = 45$

And hence the angles are 45, 90 and 45

Since one of the angle is 90 , it is right angled triangle

Since two angles are equal, two sides are equal, Hence it is isosceles triangle.

Answer 2

❍ Let the angles of the traingle be x, 2x and x respectively.

$\underline{\pink{\bigstar\:\boldsymbol{By\: Using\; ASP \; Property\::}}}$⠀⠀

Angle Sum Property of Triangle is basically, sum of all angles of the triangle is 180°.

Therefore,

$\dashrightarrow\sf x + 2x + x = 180^\circ \\\\\\\dashrightarrow\sf 2x + 2x = 180^\circ \\\\\\\dashrightarrow\sf 4x = 180^\circ \\\\\\\dashrightarrow\sf x = \cancel\dfrac{180^\circ}{4} \\\\\\\dashrightarrow{\underline{\boxed{\frak{\pink{x = 45^\circ}}}}}$

Hence,

First angle, x = 45°

Second angle, x = 45°

Third angle, 2x = 2(45)° = 90°

$\therefore{\underline{\sf{Hence,\;angles\;of\;the\;\triangle\;are\; \bf{45^\circ,90^\circ\;\&\;45^\circ}.}}}$

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C L A S S I F Y I N G :

Here, we can see that Triangle has two equal angles, (45° and 45°). Therefore, the triangle is an isosceles triangle. (Isosceles triangle has two equal angles).

And, In the given triangle one angle is equal to 90° also. Therefore, the triangle is right – angled triangle.