D. The Distance Between a Point and the Origin
Completion requirements
D. The Distance Between a Point and the Origin
Given any point \(P(x, y)\) on the terminal arm of an angle in standard position, you can construct a right triangle by drawing a vertical line from the point to the \(x\)-axis.
The height of the resulting triangle is equal to the \(y\)-coordinate of the point, and the base length of the triangle is equal to the \(x\)-coordinate. Since the lengths of the legs of the right triangle are known, you can determine the distance, \(r\), from the origin to the given point, using the Pythagorean Theorem as follows:
\[\begin{align}
r^2 &= x^2 + y^2 \\
r &= \sqrt {x^2 + y^2 } \\
\end{align}\]
The height of the resulting triangle is equal to the \(y\)-coordinate of the point, and the base length of the triangle is equal to the \(x\)-coordinate. Since the lengths of the legs of the right triangle are known, you can determine the distance, \(r\), from the origin to the given point, using the Pythagorean Theorem as follows:
\[\begin{align}
r^2 &= x^2 + y^2 \\
r &= \sqrt {x^2 + y^2 } \\
\end{align}\]
