Questions: Example 4 Suppose M is the midpoint of FG. Find each missing 34. FM=5y+13, MG=5-3y, FG=? 35. FM

Transcript text: Example 4 Suppose $M$ is the midpoint of $\overline{F G}$. Find each missing 34. $F M=5 y+13, M G=5-3 y, F G=$ ? 35. FM

Solution

Solution Steps

To solve the given problem, we need to use the property that the midpoint divides a line segment into two equal parts. This means that $ FM = MG $. We can set up an equation using the given expressions for $ FM $ and $ MG $ and solve for $ y $. Once we have $ y $, we can find the length of $ FG $ by adding $ FM $ and $ MG $.

Solution Approach

Set up the equation $ FM = MG $ using the given expressions.
Solve for $ y $.
Substitute $ y $ back into the expressions for $ FM $ and $ MG $ to find their lengths.
Add the lengths of $ FM $ and $ MG $ to find $ FG $.

Step 1: Set Up the Equation

Given that $ M $ is the midpoint of $ \overline{FG} $, we know that $ FM = MG $. The expressions for $ FM $ and $ MG $ are: \[ FM = 5y + 13 \] \[ MG = 5 - 3y \]

Step 2: Solve for $ y $

Set up the equation $ FM = MG $: \[ 5y + 13 = 5 - 3y \]

Solve for $ y $: \[ 5y + 3y = 5 - 13 \] \[ 8y = -8 \] \[ y = -1 \]

Step 3: Calculate $ FM $ and $ MG $

Substitute $ y = -1 $ back into the expressions for $ FM $ and $ MG $: \[ FM = 5(-1) + 13 = -5 + 13 = 8 \] \[ MG = 5 - 3(-1) = 5 + 3 = 8 \]

Step 4: Calculate $ FG $

Since $ FG = FM + MG $: \[ FG = 8 + 8 = 16 \]