Given EP = FP and GQ = FQ, what is the perimeter of ΔEFG?

Steps to Solve

1. Identify the lengths of the sides

The lengths of the sides of triangle ( \Delta EFG ) are given as:

• ( EF = 2x )
• ( FG = 3x - 1 )
• ( EG = 4y + 2 + 4y + 4 = 8y + 6 )

2. Set up the equations using the given conditions

From the problem, since ( EP = FP ) and ( GQ = FQ ), we have:

• ( EP + FP = EF )
• ( GQ + FQ = FG )

Thus,

1. ( x + 2y = 2x )

2. ( 4y + 4 = 3x - 1 )

3. Solve the first equation

Rearranging ( x + 2y = 2x ):

$$ 2y = 2x - x \implies 2y = x \implies x = 2y $$

4. Solve the second equation

Now substituting ( x = 2y ) into ( 4y + 4 = 3x - 1 ):

$$ 4y + 4 = 3(2y) - 1 $$

This simplifies to:

$$ 4y + 4 = 6y - 1 $$

Rearranging gives:

$$ 4y - 6y = -1 - 4 \implies -2y = -5 \implies y = \frac{5}{2} $$

5. Find ( x ) using ( y )

Substituting ( y = \frac{5}{2} ) back to find ( x ):

$$ x = 2y = 2 \left( \frac{5}{2} \right) = 5 $$

6. Calculate the lengths of the sides

Now substituting ( x = 5 ) and ( y = \frac{5}{2} ) into the expressions for the sides:

• ( EF = 2x = 2(5) = 10 )
• ( FG = 3x - 1 = 3(5) - 1 = 15 - 1 = 14 )
• ( EG = 8y + 6 = 8\left( \frac{5}{2} \right) + 6 = 20 + 6 = 26 )

7. Calculate the perimeter

The perimeter ( P ) of triangle ( \Delta EFG ) is given by:

$$ P = EF + FG + EG = 10 + 14 + 26 = 50 $$