Questions: R is the midpoint between Q and L. If QR=4x-2 and QL=5x+11, find the length of RL?

R is the midpoint between Q and L. If QR=4x-2 and QL=5x+11, find the length of RL?

Transcript text: 1) $R$ is the midpoint between $Q$ and $L$. If $Q R=4 x-2$ and $Q L=5 x+11$, find the length of $R L$ ? (**HINT: Draw a picture ${ }^{* *}$ ) (3 points)

Solution

Solution Steps

To find the length of \( RL \), we need to use the fact that \( R \) is the midpoint of \( QL \). This means that \( QR = RL \) and \( QL = QR + RL \). Given \( QR = 4x - 2 \) and \( QL = 5x + 11 \), we can set up an equation to solve for \( x \) and then find \( RL \).

Step 1: Set Up the Equations

Given:

\( QR = 4x - 2 \)
\( QL = 5x + 11 \)

Since \( R \) is the midpoint of \( QL \), we have: \[ QL = 2 \times QR \]

Step 2: Formulate the Equation

Substitute the given expressions into the equation: \[ 5x + 11 = 2 \times (4x - 2) \]

Step 3: Solve for \( x \)

Simplify and solve the equation: \[ 5x + 11 = 8x - 4 \] \[ 11 + 4 = 8x - 5x \] \[ 15 = 3x \] \[ x = 5 \]

Step 4: Calculate \( RL \)

Since \( QR = RL \), substitute \( x = 5 \) into the expression for \( QR \): \[ QR = 4x - 2 \] \[ QR = 4(5) - 2 \] \[ QR = 20 - 2 \] \[ QR = 18 \]

Thus, \( RL = 18 \).

Final Answer

\[ \boxed{RL = 18} \]

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