Question

find tu and ru in parallelogram rstu.

Explanation:

Step1: Use property of parallelogram

In a parallelogram, opposite - sides are equal. So, $RU = ST$ and $TU=RS$. We set up the equation based on the equality of opposite - sides. Let's assume $RU = ST$. We have $10v + 36=v + 33$ (this is wrong, we should use $RU$ and $ST$ not in this way. The correct is $RU$ and $ST$ are not the pair we should use first. We use $TU = RS$). So, $v + 33=8v+46$.

Step2: Solve the equation for $v$

Subtract $v$ from both sides: $33=8v - v+46$, which simplifies to $33 = 7v+46$. Then subtract 46 from both sides: $33 - 46=7v$, so $- 13 = 7v$. Then $v=-\frac{13}{7}$. This is wrong. Let's start over. Since opposite sides of a parallelogram are equal, we set $RU = ST$ and $TU = RS$. The correct equation from the opposite - side property is $v + 33=8v+46$. Rearranging gives $8v - v=33 - 46$, so $7v=-13$, $v = - \frac{13}{7}$. This is wrong. The correct pair: $RU$ and $ST$ are not equal in this way. We should use $TU$ (opposite to $RS$). So, $v + 33=8v+46$ is wrong. The correct is $10v+36 = 8v + 46$.

Step3: Solve the correct equation for $v$

Subtract $8v$ from both sides: $10v-8v + 36=8v-8v + 46$, which gives $2v+36 = 46$. Then subtract 36 from both sides: $2v=46 - 36$, so $2v = 10$. Divide both sides by 2: $v = 5$.

Step4: Find the length of $TU$

Substitute $v = 5$ into the expression for $TU$. Since $TU=v + 33$, then $TU=5 + 33=38$.

Step5: Find the length of $RU$

Substitute $v = 5$ into the expression for $RU$. Since $RU=10v+36$, then $RU=10\times5+36=50 + 36=86$.

Answer:

$TU = 38$
$RU = 86$