Question

A right triangle is formed in the first quadrant by the \(x-\) and \(y\) -axes and a line through the point (3,2) . Write the length \(L\) of the hypotenuse as a function of \(x\).

Short answer

The length of the hypotenuse L as a function of x is \( \frac{x\sqrt{13}}{3} \).

Step-by-step solution

Identify the coordinates

First identify the coordinates of the points that create the triangle. The base of the triangle is the line from the origin (0,0) to the point (x,0) on the x-axis, and the height is the line from the origin to the point (0,y) on the y-axis. The hypotenuse of the triangle is the line from the origin to the point (x,y).

Equation of the line

Find the equation of the line that passes through the origin and the point (3,2). Use the formula for the slope (rise/run) to get \(m = y/x = 2/3\). The equation of the line is then \(y = mx = 2x/3.\)

Hypotenuse length as a function of x

Using the Pythagorean theorem to find the length of the hypotenuse \(L\): \[L = \sqrt{x^2 + y^2} = \sqrt{x^2 + (2x/3)^2} = \sqrt{x^2 + 4x^2/9} = x\sqrt{1+4/9} = x\sqrt{13}/3.\] So, the length of the hypotenuse \(L\) as a function of \(x\) is \(L=\frac{x\sqrt{13}}{3}\).