Posted by **ej** on Sunday, March 17, 2013 at 10:18am.

a right triangle is formed in the first quadrant by the x- and y- axes and a line through the point (1,2). write the length of the hypotenuse as a function of x. find the vertices of the triangle such that its area is a minimum.

- calculus -
**Reiny**, Sunday, March 17, 2013 at 10:49am
let P(x,0) be the x-intercept and Q(0,y) be the y-intercept

label (1,2) as A(1,2)

the slope AP = slope AQ

-2/(x-1) = (2-y)/1

2x - xy - 2 + y = -2

y(1-x) = -2x

y = -2x/(1-x) or 2x/(x-1)

H^2 = x^2 + y^2

= x^2 + ( (2x/(x-1) )^2 --- simplify if needed

(don't know why we are finding the hypotenuse ? )

Area of triangle OPQ

= (1/2)base x height

= (1/2)xy

= (1/2)x(2x/(x-1))

= (1/2) (2x^2/(x-1) )

dA/dx = (1/2) [ (x-1)(4x) - 2x^2(1) ]/(x-1)^2

= (1/2) [ 4x^2 - 4x - 2x^2 ]/(x-1)^2

= 0 for a max of A

2x^2 - 4x = 0

x(2x-4) = 0

x=0 ---- > makes no sense, look at diagram

or

x = 2

then y = 2(2)/(2-1) = 4

the triangle has (2,) and (0, 4) as its vertices for a maximum area

- calculus -
**Reiny**, Sunday, March 17, 2013 at 10:51am
sticky keyboard, last line should obviously say

the triangle has (2,0) and (0, 4) as its vertices for a maximum area

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