Problem

Test the claim that triangle \(XYZ\) with \(X(0,0)\), \(Y(2,1)\), \(Z(5,0)\) is isosceles. Calculate \(\text{all}~\text{three}~\text{squared}~\text{side}~\text{lengths}\), assess the equal-side condition, and classify the triangle.

Source coordinate plane: Triangle XYZ has vertices X(0,0), Y(2,1), and Z(5,0); no side lengths or classification are shown.

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claim = true; measurements = XY²=5, XZ²=25, YZ²=10; condition assessment = X and Z share y=0, so the other sides are equal; conclusion = △XYZ is isosceles

claim = false; measurements = XY²=5, XZ²=25, YZ²=10; condition assessment = XZ is longest, so no two sides can be equal; conclusion = △XYZ is scalene

claim = true; measurements = XY²=5, XZ²=25, YZ²=5; condition assessment = XY and YZ are equal; conclusion = △XYZ is isosceles

claim = false; measurements = XY²=5, XZ²=25, YZ²=10; violated condition = no two side lengths are equal; conclusion = △XYZ is scalene, not isosceles

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Test the claim that triangle \(XYZ\) with \(X(0,0)\), \(Y(2,1)\), \(Z(5,0)\) is isosceles. Calculate \(\text{all}~\text{three}~\text{squared}~\text{side}~\text{lengths}\), assess the equal-side condition, and classify the triangle.

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