To find the perimeter of the right isosceles triangle, we start by identifying the vertices of the triangle. The right-angle vertex is at (4, 2), and one of the other vertices is at (8, 2).
Since the triangle is isosceles and right-angled, the two legs of the triangle will be equal. The distance from (4, 2) to (8, 2) is one leg of the triangle.
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Calculating the length of one leg: \[ \text{Distance} = x_2 - x_1 = 8 - 4 = 4 \text{ units} \] So one leg is 4 units long.
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Since this is a right isosceles triangle, the other leg will also be 4 units long.
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Finding the length of the hypotenuse: The hypotenuse can be calculated using the Pythagorean theorem: \[ c = \sqrt{a^2 + b^2} = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
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Calculating the perimeter: The perimeter \( P \) of the triangle is: \[ P = \text{leg}_1 + \text{leg}_2 + \text{hypotenuse} = 4 + 4 + 4\sqrt{2} \]
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Calculating \( 4\sqrt{2} \): The approximate value of \( \sqrt{2} \) is about 1.414. Thus: \[ 4\sqrt{2} \approx 4 \times 1.414 \approx 5.656 \]
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Now summing the lengths: \[ P \approx 4 + 4 + 5.656 \approx 13.656 \]
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Rounding to the nearest tenths place: \[ P \approx 13.7 \]
Thus, the perimeter of the triangle is approximately 13.7 units.
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