1. The problem asks to find the simplified expression for the perimeter of a triangle with side lengths given by the expressions: Left side: $9x - 6w$ Upper side: $2x + 9y$ Bottom side: $4y + 2w$ 2. The perimeter of a triangle is the sum of the lengths of all its sides. So, the perimeter $P$ is: $$P = (9x - 6w) + (2x + 9y) + (4y + 2w)$$ 3. Combine like terms by grouping coefficients of $x$, $y$, and $w$: $$P = (9x + 2x) + (9y + 4y) + (-6w + 2w)$$ 4. Simplify each group: $$P = 11x + 13y - 4w$$ 5. So, the simplified expression for the perimeter is: $$\boxed{11x + 13y - 4w}$$ --- 6. Next, we analyze the four given expressions to see how many are equal: 1) $2w + 9x + 13y + 8w + 4x - 6y$ 2) $7w + 7y + 7x + 6x - 14y + 4w$ 3) $11x - 2y + 12w - 5y - 2w + 2x$ 4) $4w - 3x - y + 6w + 16x - 6y$ 7. Simplify each expression by combining like terms: 1) Combine $w$: $2w + 8w = 10w$ Combine $x$: $9x + 4x = 13x$ Combine $y$: $13y - 6y = 7y$ So expression 1 is: $13x + 7y + 10w$ 2) Combine $w$: $7w + 4w = 11w$ Combine $x$: $7x + 6x = 13x$ Combine $y$: $7y - 14y = -7y$ So expression 2 is: $11w + 13x - 7y$ 3) Combine $x$: $11x + 2x = 13x$ Combine $y$: $-2y - 5y = -7y$ Combine $w$: $12w - 2w = 10w$ So expression 3 is: $13x - 7y + 10w$ 4) Combine $w$: $4w + 6w = 10w$ Combine $x$: $-3x + 16x = 13x$ Combine $y$: $-y - 6y = -7y$ So expression 4 is: $10w + 13x - 7y$ 8. Now compare the simplified expressions: - Expression 1: $13x + 7y + 10w$ - Expression 2: $11w + 13x - 7y$ - Expression 3: $13x - 7y + 10w$ - Expression 4: $10w + 13x - 7y$ 9. Expressions 3 and 4 are identical (order of terms does not matter): $13x - 7y + 10w$ 10. Expression 2 differs in the coefficient of $w$ (11w vs 10w) and expression 1 differs in the sign of $y$ term. 11. Therefore, only expressions 3 and 4 are equal. --- **Final answers:** - Simplified perimeter: $11x + 13y - 4w$ - Number of equal expressions among the four given: 2 (expressions 3 and 4)