Question
In a $\triangle\text{ABC},\angle\text{A}=\text{x}^\circ,\angle\text{B}=3\text{x}^\circ$ and $\angle\text{C}=\text{y}^\circ$ if $3y - 5x = 30$, , prove that the triangle is right angled.
✓
Answer
We have,
$\angle\text{A}=\text{x}^\circ\ ....(\text{i})$
$\angle\text{B}=3\text{x}^\circ\ .....(\text{ii})$
And, $\angle\text{C}=\text{y}^\circ\ ......(\text{iii})$
We know that, the sum of angles of a triangle is $180^\circ $
$\therefore\angle\text{A}+\angle\text{B}+\angle\text{C}=180^\circ$
$x + 3x + y = 180$ [using $(i), (ii)$ and $(iii)]$
$\Rightarrow 4x + y = 180 .....(iv)$
Now, 3$y - 5x = 30 .....(v)$ [given]
Multiplying equation $(iv)$ by $3$ we get
$12x + 3y = 540 .....(vi)$
Subtracting equation $(v)$ from equation $(vi)$ we get
$12x + 5x = 540 - 30$
$\Rightarrow 17x = 510$
$\Rightarrow\text{x}=\frac{510}{17}$
$\Rightarrow x = 30$
Putting $x = 30$ in equation $(iv)$ we get
$4 \times 30 + y = 180$
$\Rightarrow 120 + y = 180$
$\Rightarrow y = 180 - 120$
$\Rightarrow y = 60$
Now, $\angle\text{B}=3\text{x}^\circ$
$\Rightarrow\angle\text{B}=3\times30^\circ$
$\Rightarrow\angle\text{B}=90^\circ$
$\therefore\triangle\text{ABC}$ is the right angle triangle.
Hence prived.
$\angle\text{A}=\text{x}^\circ\ ....(\text{i})$
$\angle\text{B}=3\text{x}^\circ\ .....(\text{ii})$
And, $\angle\text{C}=\text{y}^\circ\ ......(\text{iii})$
We know that, the sum of angles of a triangle is $180^\circ $
$\therefore\angle\text{A}+\angle\text{B}+\angle\text{C}=180^\circ$
$x + 3x + y = 180$ [using $(i), (ii)$ and $(iii)]$
$\Rightarrow 4x + y = 180 .....(iv)$
Now, 3$y - 5x = 30 .....(v)$ [given]
Multiplying equation $(iv)$ by $3$ we get
$12x + 3y = 540 .....(vi)$
Subtracting equation $(v)$ from equation $(vi)$ we get
$12x + 5x = 540 - 30$
$\Rightarrow 17x = 510$
$\Rightarrow\text{x}=\frac{510}{17}$
$\Rightarrow x = 30$
Putting $x = 30$ in equation $(iv)$ we get
$4 \times 30 + y = 180$
$\Rightarrow 120 + y = 180$
$\Rightarrow y = 180 - 120$
$\Rightarrow y = 60$
Now, $\angle\text{B}=3\text{x}^\circ$
$\Rightarrow\angle\text{B}=3\times30^\circ$
$\Rightarrow\angle\text{B}=90^\circ$
$\therefore\triangle\text{ABC}$ is the right angle triangle.
Hence prived.
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