Respuesta :
To find the value of ( y ), we'll use the fact that the sum of the lengths of all three segments ( RS ), ( ST ), and ( RT ) equals 66.Given:( RS = 8y + 5 )( ST = 5y + 9 )( RT = 66 )We'll first substitute the given values into the equation for ( RT ) and solve for ( y ):[ RS + ST + RT = 66 ]Substitute the given expressions for ( RS ) and ( ST ): [ (8y + 5) + (5y + 9) + 66 = 66 ]Combine like terms: [ 8y + 5 + 5y + 9 + 66 = 66 ] [ 13y + 80 = 66 ]Subtract 80 from both sides: [ 13y = -14 ]Now, divide both sides by 13 to isolate ( y ): [ y = \frac{-14}{13} ]Now that we have found the value of ( y ), we can find the lengths of ( RS ) and ( ST ) by substituting ( y ) into their respective expressions:[ RS = 8y + 5 = 8\left(\frac{-14}{13}\right) + 5 ] [ RS = -\frac{112}{13} + 5 = -\frac{112}{13} + \frac{65}{13} = \frac{-112 + 65}{13} = \frac{-47}{13} ][ ST = 5y + 9 = 5\left(\frac{-14}{13}\right) + 9 ] [ ST = -\frac{70}{13} + 9 = -\frac{70}{13} + \frac{117}{13} = \frac{-70 + 117}{13} = \frac{47}{13} ]So, the value of ( y ) is ( -\frac{14}{13} ), ( RS ) is ( -\frac{47}{13} ), and ( ST ) is ( \frac{47}{13} ).
