So I have this major assignment due, and I cant seem to figure this one out:
A right triangle is formed in the first quadrant by the x- and y-axes and a line through the point (5, 3). Write the length L of the hypotenuse as a function of x(the x-intercept of the line).
Can someone please help me on this problem? The answer has to be in the most simplified version possible. An explanation would greatly be appreciated. thank you
Make a sketch
let the x-intercept be (x,0) and the y-intercept (0,y)
Draw an altitude from (5,3) to the x-axis
I see two similar right-angled triangles
By ratios:
y/x = 3/(x-5)
y = 3x/(x-5)
L^2 = x^2 + y^2
= x^2 + 9x^2/(x-5)^2
L = √( x^2 + 9x^2/(x-5)^2 )
= √[ (x^2(x-5)^2 + 9x^2)/(x-5)2 ]
= 1/(x-5) √[ x^2(x-5)^2 + 9 ]
= x/(x-5) √ (x^2 - 10x + 34 )
thanks...i got that..the problem is i cant put that into webassign..it wont take my answer....how can i further simplify this answer
Sure! I'd be happy to help.
To find the length of the hypotenuse (denoted by L) as a function of x, we need to use the Pythagorean theorem. The Pythagorean theorem states that for any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, the other two sides are the distances from the point (5, 3) to the x-axis and the y-axis.
Let's call the distance from (5, 3) to the x-axis d₁ and the distance from (5, 3) to the y-axis d₂.
Since the x-axis is perpendicular to the line passing through (5, 3), the distance d₁ is simply the y-coordinate of the point (5, 3), which is 3.
Similarly, since the y-axis is perpendicular to the line, the distance d₂ is the x-coordinate of the point (5, 3), which is 5.
Now, using the Pythagorean theorem, we can write the equation:
L² = d₁² + d₂²
Substituting the values we found for d₁ and d₂, the equation becomes:
L² = 3² + 5²
L² = 9 + 25
L² = 34
To find L, we take the square root of both sides of the equation:
L = √34
So the length of the hypotenuse, L, as a function of x is:
L(x) = √34
And that's the simplified version of the function.