Question

In tringle ABC angle a is x b is 3x c is y if 3y-5x= 30 prove that it is a right angle triangle

Answer 1

$since \: the \: sum \: of \: all \: angles \: in \: a \: triangle \: = \: 180 \\ \\ therefore \: a + b + c = 180 \\ \\ thus \: x + 3x + y = 180 \\ \\ or \: \: 4x + y = 180 \\ \\ y = 180 - 4x...........(1) \\ \\ given \: that \: \: 3y - 5x = 30......(2) \\ \\ putting \: the \: value \: of \: y \: from \: equation \: (1) \: in \: equation \: (2) \\ \\ so \: \: 3(180 - 4x) - 5x = 30 \\ \\ 540 - 12x - 5x = 30 \\ \\ - 17x = 30 - 540 \\ \\ - 17x = - 510 \\ \\ x = \frac{510}{17} \\ \\ x = 30 \\ \\ since \: angle \: c = 3x \\ \\ therefore \: angle \: c \: = 30 \times 30 = 90 \\ \\ hence \: given \: triangle \: is \: a \: right \: angle \: triangle \\ \\ \: \: \: \: \: \: \: proved \: \: \: \: \\ \\ hope \: it \: helps$

Answer 2

Answer:

given :

∠A = x°, ∠B = 3x° and ∠C = y°.

∠A + ∠B + ∠C = 180° (by angle sum property of a triangle)

⇒ x + 3x + y = 180°

⇒ 4x + y = 180° …..(i)

Also 5x – 3y + 30 = 0

⇒ 5x – 3y = -30 …(ii)

Multiply equation (i) by 3 to make coecient of y equal and then adding with (ii), we get

17x = 510 ⇒ x = 30°

Now putting the value of x in equation (i), we get

4 x 30° + y – 180°

y = 180° – 120°

y = 60°

Sum of two angles x and y is 90°

Hence, it is a right angled triangle.