Question

Calculate the length of a diagonal from the origin to the point \( (6, 8) \).

Ask by Todd Nguyen. in South Africa
Jan 22,2025

Real Tutor Solution

Tutor-Verified Answer

Answer

The length of the diagonal from the origin to the point (6, 8) is 10 units.

Solution

To calculate the length of the diagonal (also known as the distance) from the origin \((0, 0)\) to the point \((6, 8)\) in a Cartesian coordinate system, you can use the **distance formula**. The distance formula is derived from the Pythagorean theorem and is given by: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] **Step-by-Step Calculation:** 1. **Identify the coordinates:** - Origin: \((x_1, y_1) = (0, 0)\) - Given point: \((x_2, y_2) = (6, 8)\) 2. **Plug the values into the distance formula:** \[ \text{Distance} = \sqrt{(6 - 0)^2 + (8 - 0)^2} \] 3. **Simplify the expressions inside the parentheses:** \[ \text{Distance} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} \] 4. **Add the squared terms:** \[ \text{Distance} = \sqrt{100} \] 5. **Take the square root:** \[ \text{Distance} = 10 \] **Final Answer:** The length of the diagonal from the origin to the point \((6, 8)\) is **10 units**.

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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!

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