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Chapter 32: Problem 1038

Show that the triangle with \((-3,2),(1,1)\), and \((-4,-2)\) as vertices is an isosceles triangle.

Short Answer

Expert verified
We calculated the side lengths of the triangle with vertices \((-3,2)\), \((1,1)\), and \((-4,-2)\) as follows: \(AB = \sqrt{17}\), \(BC = \sqrt{34}\), and \(CA = \sqrt{17}\). Since \(AB = CA\), the triangle is an isosceles triangle.

Step by step solution

01

Calculate the length of side AB

To calculate the distance between points \(A(-3,2)\) and \(B(1,1)\), we can use the distance formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] Substitute the coordinates of points A and B into the formula: \[AB = \sqrt{(1 - (-3))^2 + (1 - 2)^2} = \sqrt{(4)^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17}\]
02

Calculate the length of side BC

Next, we calculate the distance between points \(B(1,1)\) and \(C(-4,-2)\): \[BC = \sqrt{(-4 - 1)^2 + (-2 - 1)^2} = \sqrt{(-5)^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34}\]
03

Calculate the length of side CA

Finally, we calculate the distance between points \(C(-4,-2)\) and \(A(-3,2)\): \[CA = \sqrt{(-3 + 4)^2 + (2 + 2)^2} = \sqrt{(1)^2 + (4)^2} = \sqrt{1 + 16} = \sqrt{17}\]
04

Check for equal side lengths

Now that we have the lengths of all three sides of the triangle, we can check for equal side lengths: - Side AB: \(\sqrt{17}\) - Side BC: \(\sqrt{34}\) - Side CA: \(\sqrt{17}\) Since \(AB = CA\), we can conclude that the triangle with vertices \((-3,2)\), \((1,1)\), and \((-4,-2)\) is an isosceles triangle.

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