Question

in triangle ABC, angle a= x degree , angle b=3x , angle c= y degree . If 3y-5x=30,prove that the triangle is a right triangle

Answer 1

∠A+∠B+∠C= 180°
x+3x+y=180°
4x+y=180°
operating 3×4x+y-3y-5x=3×180-30°
12x+5x=540°-30°
17x=510°
x=30°
∠A=30
∴∠B=3x=3×30=90
HENCE IT IS A RIGHT ANGLE

Answer 2

Answer:

The given triangle is a right angled triangle as it contains one of its angle as 90°

Solution:

We know that sum of all the angles of a triangle = 180°

Here

$\angle A = x ^ { \circ } , \angle B = 3 x ^ { \circ } , \angle C = y ^ { \circ }$

Thus,

$\begin{array} { c } { \angle A + \angle B + \angle C = 180 ^ { \circ } } \\\\ { x ^ { \circ } + 3 x ^ { \circ } + y ^ { \circ } = 180 ^ { \circ } } \\\\ { 4 x ^ { \circ } + y ^ { \circ } = 180 ^ { \circ } } \\\\ { y ^ { \circ } = 180 ^ { \circ } - 4 x ^ { \circ } \ldots ( i ) } \end{array}$

Given,

$\begin{array} { l } { 3 y ^ { \circ } - 5 x ^ { \circ } = 30 ^ { \circ } } \\\\ { 3 y ^ { \circ } = 30 ^ { \circ } + 5 x ^ { \circ } \quad \ldots . ( i i ) } \end{array}$

Hence, to put the value of 3° in equation (i), multiply equation (i) by 3 on both the sides

$\begin{array} { c } { 3 y ^ { \circ } = 3 \left( 180 - 4 x ^ { \circ } \right) } \\\\ { 30 ^ { \circ } + 5 x ^ { \circ } = 3 \left( 180 - 4 x ^ { \circ } \right) } \\\\ { 30 ^ { \circ } + 5 x ^ { \circ } = 540 ^ { \circ } - 12 x ^ { \circ } } \\\\ { 5 x ^ { \circ } + 12 x ^ { \circ } = 540 ^ { \circ } - 30 ^ { \circ } } \\\\ { 17 x ^ { \circ } = 510 ^ { \circ } } \\\\ { x ^ { \circ } = \frac { 510 ^ { \circ } } { 17 } = 30 ^ { \circ } } \end{array}$

$y ^ { \circ } = 180 ^ { \circ } - 4 x ^ { \circ } = 180 ^ { \circ } - 4 \times 30 ^ { \circ } = 180 ^ { \circ } - 120 ^ { \circ } = 60 ^ { \circ }$

∴ The three angles of the triangle are:

$\begin{aligned} \angle A & = x ^ { \circ } = 30 ^ { \circ } \\\\ \angle B = 3 x ^ { \circ } & = 3 \times 30 ^ { \circ } = 90 ^ { \circ } \\\\ \angle C & = y ^ { \circ } = 60 ^ { \circ } \end{aligned}$

Here, one of the angle of the triangle is 90°. Hence, the given triangle is a right angled triangle.