Question

The angles in a right triangle are 5x-3 , 9x , 3x+13 , show that the triangle is right angled

Answer 1

It’s given, the angles in a right triangle are 5x-3 , 9x , 3x+13 . We have asked to prove that the given triangle is a right-angled triangle.

• Proof:-

`\small \underline{ \boxed{ \sf{ \bigg(5x-3\bigg)°+\bigg(9x\bigg)°+\bigg(3x+13\bigg)°= 180° }}}\\`

`\sf \because\underline{ \:The\: three \:interior \:angles \:of \:any \:triangle \:add \:up \:to\: \red{180°}}\\`

`\:\:\:\:\:\:\longrightarrow \sf { \bigg(17x +10 \bigg)°= 180°}\\`

`\:\:\:\:\:\:\longrightarrow \sf { \bigg(17x\bigg)°= 180°-10°}\\`

`\:\:\:\:\:\:\longrightarrow \sf { \bigg(17x\bigg)°= 170°}\\`

`\:\:\:\:\:\:\longrightarrow \sf {x = (170°)/(17°)}\\`

`\:\:\:\:\:\:\longrightarrow \sf {x = (17°* 10°)/(17°)}\\`

`\:\:\:\:\:\:\longrightarrow \sf {x = \frac{\cancel{17°}* 10°}{\cancel{17°}}}\\`

`\:\:\:\:\:\:\longrightarrow \boxed{ \tt{ \pmb{ \red{x =10°}}}}\\`

`\underline{\rm{\sf 1st\:Angle:-}}`

`\sf\longrightarrow \bigg(5x-3\bigg)° = \bigg(5* 10 -3 \bigg)°=\underline{ \pink{47°}}\\`

`\underline{\rm{\sf 2nd\:Angle:-}}`

`\sf \longrightarrow 9x° =\bigg( 9* 10\bigg)° =\boxed{\underline{\pink{90°}}}\\`

`\underline{\rm{\sf 3rd\:Angle:-}}`

`\sf \longrightarrow \bigg(3x+3\bigg)° =\bigg( 3* 10+13 \bigg)°= \underline{\pink{43°}}\\`

Since, one of the angles is 90°,henceforth we can say that the given triangle is a right-angled triangle.

`\:\:\:\:\:\:\\ \underline{ \cal{ \pmb{ \: \frak{\purple{Proved! \: }. }}}}\\`