To find the width of the rectangle, you can use the formula for the area of a rectangle:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given that the area is \(x^2 + x - 12\) and the length is \(x + 4\), you can set up the equation:
\[ x^2 + x - 12 = (x + 4) \times \text{Width} \]
To find the width, divide the area by the length:
\[ \text{Width} = \frac{x^2 + x - 12}{x + 4} \]
Now, we can perform polynomial long division or simplify it:
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Factor \(x^2 + x - 12\): \[ x^2 + x - 12 = (x - 3)(x + 4) \]
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Therefore, we can substitute back in: \[ \frac{(x - 3)(x + 4)}{x + 4} \]
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This simplifies to: \[ x - 3 \]
So, the width of the rectangle is \(x - 3\).
Next, for the triangles:
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Perimeter of the first triangle (9, 12):
- Use the Pythagorean theorem to find the hypotenuse \(c_1\): \[ c_1 = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15 \]
- So, the perimeter \(P_1\) is: \[ P_1 = 9 + 12 + 15 = 36 \]
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Perimeter of the second triangle (12, 16):
- Again using the Pythagorean theorem for hypotenuse \(c_2\): \[ c_2 = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20 \]
- Therefore, the perimeter \(P_2\) is: \[ P_2 = 12 + 16 + 20 = 48 \]
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Now, find how much longer the perimeter of the larger triangle is than that of the smaller triangle: \[ P_2 - P_1 = 48 - 36 = 12 \]
Thus, the perimeter of the larger triangle is 12 units longer than that of the smaller triangle.
Final Answers:
- Width of the rectangle: \(x - 3\)
- Difference in perimeter lengths: \(12\)
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