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

Steps to Solve

1. Set relationships between segments

Since ( EP = FP = x + 2y ) and ( GQ = FQ = 4y + 4 ), we can write:

$$ EP = x + 2y $$ $$ FP = x + 2y $$ $$ GQ = 4y + 4 $$ $$ FQ = 4y + 4 $$

2. Identify the triangle's sides

From the triangle sides given:

• Side ( EF = 4y + 2 )
• Side ( FG = 3x - 1 )
• Side ( GE = 2x )

3. Express the perimeter

The perimeter ( P ) of triangle ( EFG ) can be found by adding the lengths of the sides:

$$ P = EF + FG + GE $$ Substituting the expressions for these sides:

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

4. Combine like terms

Now we combine the terms:

$$ P = (4y + 2) + (3x + 2x - 1) $$ $$ P = 4y + 2 + 5x - 1 $$ $$ P = 5x + 4y + 1 $$

5. Set the perimeter equal to 42

We know that the perimeter is 42:

$$ 5x + 4y + 1 = 42 $$

6. Solve for ( x ) and ( y )

Now, we simplify this equation:

$$ 5x + 4y = 41 $$

To solve, we need either another equation or values for ( x ) or ( y ). Typically, we might assume values or further relationships between ( x ) and ( y ) based on the triangle's context.

Assuming more value relationships, we analyze them to find specific ( x ) and ( y ) values that satisfy ( 5x + 4y = 41 ).

7. Identify ( x ) and ( y )

Let’s try ( x = 5 ) and solve for ( y ):

$$ 5(5) + 4y = 41 $$ $$ 25 + 4y = 41 $$ $$ 4y = 16 $$ $$ y = 4 $$

Now substituting back to find the actual lengths of the sides:

• ( EF = 4y + 2 = 4(4) + 2 = 18 )
• ( FG = 3x - 1 = 3(5) - 1 = 14 )
• ( GE = 2x = 2(5) = 10 )

8. Calculate the final perimeter

Now, we calculate the perimeter:

$$ P = 18 + 14 + 10 = 42 $$