A right triangle is formed in the first quadrant by the x- and y-axes and a line through the point (8,4).

Write the length L of the hypotenuse as a function of x.

Completely stuck came up with the sqrt(x^2+Y^2)=L, not sure what to do? can anyone guide me in the right direction?

any line through (8,4) in the first quadrant is

y-4 = m(x-8)

So, if the line in question has slope m, the x-intercept is 8-(4/m) and the y-intercept is 4-8m

So,

L = √x^2+y^2 = √((8-(4/m))^2 + (4-8m)^2)
= -4/m √((m^2+1)(2m-1)^2)

Of course, there are other ways to characterize the line, but something needs to be specified besides just a point on it.

Check: If the slope is, say, -1, the intercepts are both 12, so we know L = 12√2.
Our formula says L=12√2
So, it checks out at at least one point.

To find the length L of the hypotenuse as a function of x, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). In this case, the x-axis and y-axis form the two sides, and the line through the point (8,4) represents the hypotenuse.

Let's consider the point (8,4). We can see that the triangle formed by the x-axis, y-axis, and the line passing through (8,4) is a right triangle. The length of the side adjacent to the x-axis can be found by measuring the horizontal distance from the origin to the point (8,4), which is simply x = 8. The length of the side adjacent to the y-axis can be found by measuring the vertical distance from the origin to the point (8,4), which is y = 4.

Now, we can apply the Pythagorean theorem:

c^2 = a^2 + b^2

L^2 = x^2 + y^2

Substituting the values x = 8 and y = 4 into the equation, we get:

L^2 = 8^2 + 4^2
L^2 = 64 + 16
L^2 = 80

To find the length L, we need to take the square root of both sides of the equation:

L = √80

Simplifying further, we have:

L = √(16 * 5)
L = 4√5

Therefore, the length L of the hypotenuse as a function of x is L(x) = 4√5.