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see below

Step-by-step explanation:

Since RT bisects QRS

1/2  QRS = TRS

1/2 ( 9x+214) = -9x+53

Multiply each side by 2

9x +214 = -18x+ 106

Add 18x from each side

9x +214+18x = 106

27x +214 = 106

Subtract 214 from each side

27x = 106-214

27x =-108

Divide by 27

27x/27 = -108/27

x = -4

TRS = -9x+53

     =-9 *-4 +53

     = 36+ 53

    =89

x is -4.

TRS is 89.

Step-by-step explanation:

So we know that QR bisects QRS. By the definition of an angle bisector, this means that the adjacent angles are equivalent. In other words:

We are told that QRS is (9x+214). QRS is the sum of QRT and TRS. Thus:

And since we now that QRT and TRS are equivalent, substitute:

Combine like terms:

Substitute them for their equations:

Now, solve for x. On the right, distribute:

Add 18x to both sides:

Subtract 214 from both sides:

Divide both sides by 27:

So, the value of x is -4.

To find TRS, substitute x=-4 back into the equation. So:

Multiply and add:

And we're done!

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Step-by-step explanation: