QuestionPoint $S$ is the midpoint of $\overline{R T}$ . Given $R S=6 y+3$ and $S T=3 y+9$ , find $y$ , $R S$ , and $R T$ .

Studdy Solution

STEP 1

Assumptions1. Point is the midpoint of $\overline{RT}$. . The length of segment $RS$ is $6y+3$.
3. The length of segment is $3y+9$ .

STEP 2

Since is the midpoint of $\overline{RT}$, the lengths of segments $RS$ and are equal. We can set up the equation $RS =$ .
$6y+ =y+9$

STEP 3

To isolate $y$ , we can subtract $3y$ from both sides of the equation.
$6y -3y =3y+9 -3y$ This simplifies to $3y +3 =9$

STEP 4

Next, subtract3 from both sides of the equation to solve for $y$ .
$3y =9 -3$ This simplifies to $3y =6$

STEP 5

Finally, divide both sides of the equation by3 to find the value of $y$ .
$y = \frac{}{3}$ So, $y =2$ .

STEP 6

Now, we can substitute $y =2$ back into the expressions for $RS$ and $$ to find their lengths.
For $RS$ $RS =6y+3 =6(2) +3$

STEP 7

Calculate the length of $RS$ .
$RS =12 +3 =15$

STEP 8

For =3y+ =3(2) +$$

STEP 9

Calculate the length of $$.
$=6 +9 =15$

STEP 10

Finally, we can find the length of $\overline{RT}$ by adding the lengths of $RS$ and $$.
$RT = RS + =15 +15$

STEP 11

Calculate the length of $RT$ .
$RT =30$ So, $y =$ , $RS =15$ , and $RT =30$ .

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QuestionPoint $S$ is the midpoint of $\overline{R T}$ . Given $R S=6 y+3$ and $S T=3 y+9$ , find $y$ , $R S$ , and $R T$ .

Studdy Solution

STEP 1

Assumptions1. Point is the midpoint of $\overline{RT}$. . The length of segment $RS$ is $6y+3$.
3. The length of segment is $3y+9$ .

STEP 2

Since is the midpoint of $\overline{RT}$, the lengths of segments $RS$ and are equal. We can set up the equation $RS =$ .
$6y+ =y+9$

STEP 3

To isolate $y$ , we can subtract $3y$ from both sides of the equation.
$6y -3y =3y+9 -3y$ This simplifies to $3y +3 =9$

STEP 4

Next, subtract3 from both sides of the equation to solve for $y$ .
$3y =9 -3$ This simplifies to $3y =6$

STEP 5

Finally, divide both sides of the equation by3 to find the value of $y$ .
$y = \frac{}{3}$ So, $y =2$ .

STEP 6

Now, we can substitute $y =2$ back into the expressions for $RS$ and $$ to find their lengths.
For $RS$ $RS =6y+3 =6(2) +3$

STEP 7

Calculate the length of $RS$ .
$RS =12 +3 =15$

STEP 8

For =3y+ =3(2) +$$

STEP 9

Calculate the length of $$.
$=6 +9 =15$

STEP 10

Finally, we can find the length of $\overline{RT}$ by adding the lengths of $RS$ and $$.
$RT = RS + =15 +15$

STEP 11

Calculate the length of $RT$ .
$RT =30$ So, $y =$ , $RS =15$ , and $RT =30$ .

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QuestionPoint SSS is the midpoint of RT‾\overline{R T}RT. Given RS=6y+3R S=6 y+3RS=6y+3 and ST=3y+9S T=3 y+9ST=3y+9, find yyy, RSR SRS, and RTR TRT.

Studdy Solution

STEP 1

Assumptions1. Point is the midpoint of $\overline{RT}$. . The length of segment $RS$ is $6y+3$.3. The length of segment is 3y+93y+93y+9.

STEP 2

Since is the midpoint of $\overline{RT}$, the lengths of segments $RS$ and are equal. We can set up the equation RS=RS =RS=.6y+=y+96y+ =y+96y+=y+9

STEP 3

To isolate yyy, we can subtract 3y3y3y from both sides of the equation.6y−3y=3y+9−3y6y -3y =3y+9 -3y6y−3y=3y+9−3yThis simplifies to3y+3=93y +3 =93y+3=9

STEP 4

Next, subtract3 from both sides of the equation to solve for yyy.3y=9−33y =9 -33y=9−3This simplifies to3y=63y =63y=6

STEP 5

Finally, divide both sides of the equation by3 to find the value of yyy.y=3y = \frac{}{3}y=3​So, y=2y =2y=2.

STEP 6

Now, we can substitute y=2y =2y=2 back into the expressions for RSRSRS and $$ to find their lengths.For RSRSRSRS=6y+3=6(2)+3RS =6y+3 =6(2) +3RS=6y+3=6(2)+3

STEP 7

Calculate the length of RSRSRS.RS=12+3=15RS =12 +3 =15RS=12+3=15

STEP 8

For =3y+ =3(2) +$$

STEP 9

Calculate the length of $$.=6+9=15 =6 +9 =15=6+9=15

STEP 10

Finally, we can find the length of RT‾\overline{RT}RT by adding the lengths of RSRSRS and $$.RT=RS+=15+15RT = RS + =15 +15RT=RS+=15+15

STEP 11

Calculate the length of RTRTRT.RT=30RT =30RT=30So, y=y =y=, RS=15RS =15RS=15, and RT=30RT =30RT=30.

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QuestionPoint $S$ is the midpoint of $\overline{R T}$ . Given $R S=6 y+3$ and $S T=3 y+9$ , find $y$ , $R S$ , and $R T$ .

Assumptions1. Point is the midpoint of $\overline{RT}$. . The length of segment $RS$ is $6y+3$.
3. The length of segment is $3y+9$ .

Since is the midpoint of $\overline{RT}$, the lengths of segments $RS$ and are equal. We can set up the equation $RS =$ .
$6y+ =y+9$

To isolate $y$ , we can subtract $3y$ from both sides of the equation.
$6y -3y =3y+9 -3y$ This simplifies to $3y +3 =9$

Next, subtract3 from both sides of the equation to solve for $y$ .
$3y =9 -3$ This simplifies to $3y =6$

Finally, divide both sides of the equation by3 to find the value of $y$ .
$y = \frac{}{3}$ So, $y =2$ .

Now, we can substitute $y =2$ back into the expressions for $RS$ and $$ to find their lengths.
For $RS$ $RS =6y+3 =6(2) +3$

Calculate the length of $RS$ .
$RS =12 +3 =15$

Calculate the length of $$.
$=6 +9 =15$

Finally, we can find the length of $\overline{RT}$ by adding the lengths of $RS$ and $$.
$RT = RS + =15 +15$

Calculate the length of $RT$ .
$RT =30$ So, $y =$ , $RS =15$ , and $RT =30$ .