1. **State the problem:** Miriam wants to tape a rectangle so that its area equals the area of a given right triangle. We need to find the perimeter of that rectangle. 2. **Write down the given dimensions:** - Triangle base = $4x+1$ - Triangle height = $2x+3$ - Rectangle length = $4x$ - Rectangle width = $x+2$ 3. **Formula for area:** - Area of triangle = $\frac{1}{2} \times \text{base} \times \text{height}$ - Area of rectangle = $\text{length} \times \text{width}$ 4. **Set areas equal:** $$\frac{1}{2} (4x+1)(2x+3) = (4x)(x+2)$$ 5. **Expand both sides:** $$\frac{1}{2} (8x^2 + 12x + 2x + 3) = 4x^2 + 8x$$ $$\frac{1}{2} (8x^2 + 14x + 3) = 4x^2 + 8x$$ 6. **Multiply both sides by 2 to clear fraction:** $$\cancel{2} \times \frac{1}{\cancel{2}} (8x^2 + 14x + 3) = 2 \times (4x^2 + 8x)$$ $$8x^2 + 14x + 3 = 8x^2 + 16x$$ 7. **Subtract $8x^2$ from both sides:** $$8x^2 + 14x + 3 - 8x^2 = 8x^2 + 16x - 8x^2$$ $$14x + 3 = 16x$$ 8. **Subtract $14x$ from both sides:** $$14x + 3 - 14x = 16x - 14x$$ $$3 = 2x$$ 9. **Solve for $x$:** $$x = \frac{3}{2} = 1.5$$ 10. **Find rectangle dimensions:** - Length = $4x = 4 \times 1.5 = 6$ - Width = $x+2 = 1.5 + 2 = 3.5$ 11. **Calculate perimeter of rectangle:** $$P = 2(\text{length} + \text{width}) = 2(6 + 3.5) = 2(9.5) = 19$$ **Final answer:** Miriam needs 19 meters of tape to create the rectangle.