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

Steps to Solve

1. Set Up the Equations From the problem, we know that ( EP = FP ) and ( GQ = FQ ). This means:

• The length of side EF is ( 2x )
• The length of side FG is ( 3x - 1 )
• The length of side EG is ( (x + 2y) + (4y + 4) = x + 6y + 4 )

2. Equate the Relevant Sides Using the information ( EP = FP ) implies: $$ 2x = x + 6y + 4 \implies 2x - x - 6y - 4 = 0 \implies x - 6y - 4 = 0 $$ This simplifies to: $$ x = 6y + 4 \quad \text{(Equation 1)} $$

Using the information ( GQ = FQ ) implies: $$ 3x - 1 = x + 2y \implies 3x - x - 2y - 1 = 0 \implies 2x - 2y - 1 = 0 $$ This simplifies to: $$ 2x - 2y = 1 \quad \text{(Equation 2)} $$

3. Substitute to Solve for Variables Substituting Equation 1 into Equation 2: $$ 2(6y + 4) - 2y = 1 $$ Expanding this: $$ 12y + 8 - 2y = 1 \implies 10y + 8 = 1 $$ Simplifying: $$ 10y = 1 - 8 \implies 10y = -7 \implies y = -0.7 $$

4. Find x Using y Now substituting ( y ) back into Equation 1 to find ( x ): $$ x = 6(-0.7) + 4 = -4.2 + 4 = -0.2 $$

5. Calculate the Lengths of the Sides Now substitute ( x ) and ( y ) into the expressions for the triangle sides:

• Length of EF: $$ 2x = 2(-0.2) = -0.4 $$
• Length of FG: $$ 3x - 1 = 3(-0.2) - 1 = -0.6 - 1 = -1.6 $$
• Length of EG: $$ x + 6y + 4 = -0.2 + 6(-0.7) + 4 = -0.2 - 4.2 + 4 = -0.4 $$

The perimeter is the sum of the sides: $$ \text{Perimeter} = EF + FG + EG = (-0.4) + (-1.6) + (-0.4) = -2.4 $$

This result is incorrect as the total length of triangle sides can't be negative; thus, there may have been an error in assumptions or calculations regarding the relationships between the sides.

6. Verify the values for Perimeter Calculation Given that the problem states the perimeter is 42, the equations may not have been aligned with the actual triangle inequalities or lengths.