Math / Geometry

QuestionIf $G$ is the midpoint of $\overline{F H}$ and $F H=6 y-2$ , find $y$ .

Studdy Solution

STEP 1

1. $G$ is the midpoint of $\overline{FH}$ .
2. The length $FH = 6y - 2$ .
3. The length $FG = 4x + 3$ .
4. The length $GH = 7x - 12$ .
5. Since $G$ is the midpoint, $FG = GH$ .

STEP 2

1. Set up the equation from the midpoint property.
2. Solve for $x$ .
3. Use the total length of $\overline{FH}$ to form another equation.
4. Solve for $y$ .

STEP 3

Since $G$ is the midpoint of $\overline{FH}$ , we know that $FG = GH$ . Set up the equation:
$4x + 3 = 7x - 12$

STEP 4

Solve for $x$ :
$4x + 3 = 7x - 12$ Subtract $4x$ from both sides:
$3 = 3x - 12$ Add 12 to both sides:
$15 = 3x$ Divide by 3:
$x = 5$

STEP 5

Use the total length of $\overline{FH}$ to form another equation. Since $FH = FG + GH$ and $FH = 6y - 2$ :
$6y - 2 = (4x + 3) + (7x - 12)$

STEP 6

Substitute $x = 5$ into the equation:
$6y - 2 = (4(5) + 3) + (7(5) - 12)$ Simplify:
$6y - 2 = (20 + 3) + (35 - 12)$ $6y - 2 = 23 + 23$ $6y - 2 = 46$

STEP 7

Solve for $y$ :
$6y - 2 = 46$ Add 2 to both sides:
$6y = 48$ Divide by 6:
$y = 8$
Solution: The value of $y$ is $8$ .

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