Question

In a $\Delta ABC,\angle A = x,\angle B = 3x$ and $\angle C = y$. If 3y – 5x = 30, then prove that the triangle is a right angled triangle.

Answer 1

Hint: Find the angles to check if the triangle is a right angled triangle, one angle must be ${90^ \circ }$. Use the angle sum property of a triangle.

Complete step-by-step answer:
Given-(a) The angles are x, 3x and y
           (b) One equation: 3y – 5x = 30….Equation(1)
We know that the sum of all angles of a triangle must be equal to ${180^\circ}$.
So,
$\angle A{\text{ + }}\angle B + \angle C = {180^\circ}$ …….(Angle Sum Property of a Triangle)
 $x + 3x + y = {180^\circ}$
$4x + y = {180^\circ}$ ……Equation(2)
Solving equations (1) and (2) by substitution method:
Substitute the value of $y = [(180{}^\circ - 4x)]$ from equation (2) in equation (1)
3(180 – 4x) - 5x = 30
$\Rightarrow$ 540 – 12x – 5x = 30
$\Rightarrow$ 17x = 510
$\Rightarrow$ x = ${30^\circ}$
Putting this value of x in $y = [(180{}^\circ - 4x)]$
On solving, we get y = ${60^ \circ }$
We have,
$\angle {\text{A = 3}}{{\text{0}}^\circ}{\text{ }}\angle {\text{B = 3x = }}{90^ \circ }{\text{ }}\angle {\text{C = }}{60^ \circ }$
One angle is ${90^ \circ }$. So, the answer is yes, the triangle is a right angled triangle.

Note: To solve such questions, the properties of a triangle should be used. Whenever, one equation is given in 2 variables, the second equation should be found using the conditions given in the question.