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QuestionFind yy given RS=6x+5R S=6x+5, ST=8x−1S T=8x-1, and TU=πy+13T U=\pi y+13 with midpoints SS and TT.

Studdy Solution

STEP 1

Assumptions1. is the midpoint of $RT$ . is the midpoint of RURU
3. The length of segment RSRS is given by 6x+56x+5
4. The length of segment $$ is given by $8x-1$
5. The length of segment $TU$ is given by $\pi y+13$

STEP 2

Since is the midpoint of $RT$, the lengths of $RS$ and are equal. We can set up the equationRS=RS =

STEP 3

Substitute the given expressions for RSRS and intotheequation into the equation6x+5 =8x-1$$

STEP 4

To solve for xx, we can first subtract 6x6x from both sides of the equation=2x−1 =2x-1

STEP 5

Next, add1 to both sides of the equation to isolate 2x2x=2x =2x

STEP 6

Finally, divide both sides of the equation by2 to solve for xxx=3x =3

STEP 7

Since is the midpoint of $RU$, the lengths of $RT$ and $TU$ are equal. We can set up the equationRT = TU$$

STEP 8

The length of RTRT is the sum of the lengths of RSRS and . Substitute the given expressions for $RS$, , and TUTU into the equationRS+=TURS + = TU

STEP 9

Substitute the value of xx into the expressions for RSRS and 6(3)+5 +8(3)- = \pi y+13$$

STEP 10

implify the left side of the equation18+5+24−=πy+1318+5 +24- = \pi y+13

STEP 11

Further simplify the left side of the equation46=πy+1346 = \pi y+13

STEP 12

Subtract from both sides of the equation to isolate πy\pi y33=πy33 = \pi y

STEP 13

Finally, divide both sides of the equation by π\pi to solve for yyy=33πy = \frac{33}{\pi}The value of yy is 33π\frac{33}{\pi}.

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