Calculate the length of a diagonal from the origin to the point \( (6, 8) \).
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To find the length of the diagonal from the origin \((0, 0)\) to the point \((6, 8)\), we can use the distance formula, which is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Here, \((x_1, y_1) = (0, 0)\) and \((x_2, y_2) = (6, 8)\). Plugging in these coordinates, we get: \[ d = \sqrt{(6 - 0)^2 + (8 - 0)^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \] Therefore, the length of the diagonal is \(10\). --- Calculating distances like this isn't just about math; it’s the foundation of so many fields! It’s how architects determine the lengths of beams, or how game developers create real-world physics in virtual spaces. Crafting virtual landscapes? Understanding these measurements can make all the difference! Thinking of distances as simple right triangles leads to many applications. For instance, many people confuse the coordinates when working with problems in the coordinate plane. Always double-check the ordering of your points; it’s easy to mix \((x, y)\) with \((y, x)\) especially in a busy environment!