Question

The projections of a line segment on the coordinate axes are \(5,6,8\). Then the length of the line segment is

• A. \(5\)
• B. \(5\sqrt{5}\)
• C. \(6\)
• D. \(6\sqrt{6}\)
• E. \(6\sqrt{5}\)

Hint:

The length of a 3D line segment can be found using the formula \(\sqrt{x^2+y^2+z^2}\), where \(x,y,z\) are projections on axes.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The projections of a line segment on the x, y, and z axes are the absolute differences in the coordinates of its endpoints. If a line segment joins points \(P(x_1, y_1, z_1)\) and \(Q(x_2, y_2, z_2)\), its projections are \(|x_2-x_1|\), \(|y_2-y_1|\), and \(|z_2-z_1|\). The length of the segment is the distance between P and Q.
Step 2: Key Formula or Approach:
Let the length of the line segment be L. Let the projections on the x, y, and z axes be \(p_x, p_y,\) and \(p_z\) respectively.
The relationship between the length and its projections is given by the 3D version of the Pythagorean theorem:
\[ L^2 = p_x^2 + p_y^2 + p_z^2 \] \[ L = \sqrt{p_x^2 + p_y^2 + p_z^2} \] Step 3: Detailed Explanation:
We are given the lengths of the projections:
\(p_x = 5\)
\(p_y = 6\)
\(p_z = 8\)
Using the formula for the length of the line segment:
\[ L = \sqrt{5^2 + 6^2 + 8^2} \] \[ L = \sqrt{25 + 36 + 64} \] \[ L = \sqrt{125} \] To simplify the square root, we can factor 125:
\[ L = \sqrt{25 \times 5} = \sqrt{25} \times \sqrt{5} = 5\sqrt{5} \] Step 4: Final Answer:
The length of the line segment is \(5\sqrt{5}\). This corresponds to option (B).