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 hypotenuse as a function of x.
Let the x-intercept of the line be (x,0)
let the y-intercept of the line be (0,y)
we have similar triangles, and by ratios ...
y/x = 3/(x-2)
xy - 2y = 3x
y(x-2) = 3x
y = 3x/(x-2)
L^2 = x^2 + y^2
= x^2 + [3x/(x-2)]^2
= x^2 + 9x^2/(x^2 - 4x + 4)
To determine the length of the hypotenuse as a function of x, we need to find the coordinates of the right triangle's vertex that is not on the axes.
Given that the line passes through the point (3,2), we can determine its slope using the following formula:
m = (y2 - y1) / (x2 - x1)
Let's substitute the known point (3,2) into the formula:
m = (2 - 0) / (3 - 0)
m = 2/3
Now, we can find the equation of the line passing through (3,2) using the slope-intercept form:
y - y1 = m(x - x1)
Substituting the values, we get:
y - 2 = (2/3)(x - 3)
y - 2 = (2/3)x - 2
y = (2/3)x
From this equation, we can see that the line intersects the x-axis at y = 0 and the y-axis at x = 0. Thus, we can easily determine the length L (hypotenuse) using the Pythagorean theorem.
Using the coordinates of the point where the line intersects the x-axis (x, y) = (x, 0), and the coordinates of the point where the line intersects the y-axis (x, y) = (0, (2/3)x), we can form a right triangle.
The length of the horizontal side (adjacent side) is given by the x-coordinate, which is x.
The length of the vertical side (opposite side) is given by the y-coordinate, which is (2/3)x.
Now, we can calculate the length of the hypotenuse using the Pythagorean theorem:
L^2 = (x)^2 + ((2/3)x)^2
Simplifying the equation:
L^2 = x^2 + (4/9)x^2
L^2 = (13/9)x^2
Finally, we can find the length L of the hypotenuse as a function of x by taking the square root:
L = sqrt((13/9)x^2)
L = (sqrt(13)/3)x
To find the length of the hypotenuse, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
First, let's label the triangle as follows: the x-axis is the base, the y-axis is the height, and the line through the point (3,2) is the hypotenuse.
Since the triangle is formed by the x- and y-axes, the base of the triangle would be the x-coordinate of the point (3,2), which is 3. The height of the triangle would be the y-coordinate of the point (3,2), which is 2. Let's call the hypotenuse L.
According to the Pythagorean theorem, we can write the equation as:
L^2 = 3^2 + 2^2
Simplifying:
L^2 = 9 + 4
L^2 = 13
To solve for L, we take the square root of both sides:
L = √13
Therefore, the length of the hypotenuse, denoted as L, can be written as a function of x as L(x) = √13.