SOLUTION: Find the dimensions and area of the largest rectangle that can be inscribed in a right angle triangle, whose sides have lengths of 9cm, 12cm and a hypotenuse of 15cm.

Question 1043677: Find the dimensions and area of the largest rectangle that can be inscribed in a right angle triangle, whose sides have lengths of 9cm, 12cm and a hypotenuse of 15cm.
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
Find the dimensions and area of the largest rectangle that can be inscribed in a right angle triangle, whose sides have lengths of 9cm, 12cm and a hypotenuse of 15cm.
:
let x = the width of the rectangle
let y = the length
:
Draw out a right triangle with the rectangle inscribed therein, x by y
the height of the right triangle = 12 and the base = 9
:
We are interested in the small right triangles adjacent the inscribed rectangle
The tangent of the top angle = 9/12 which = of the small right triangle on top
Cross multiply
12x = 9(12-y)
12x = 108 - 9y
x = 9 - y
similarly, the tangent of the base angle = 12/9 =
Cross multiply
9y = 12(9-x)
9y = 108 - 12x
y = 12 - x
:
Area = xy
replace y with (12-x)
A = x(12-x)
A = 12x - x^2
A quadratic equation, max occurs at the axis of symmetry; x=-b/(2a)
x =
x =
x = +4.5 cm is the width of the rectangle with max area
:
Find y
y = 12 - *4.5
y = 12 - 6
y = 6 cm is length
:
Max area: 6 * 4.5 = 27 sq/cm
:
NOt sure if I explained this clearly, let me know. CK
RELATED QUESTIONS
I would appreciate your help, What are the sides of the rectangle with the largest area... (answered by scott8148)

Find the dimension of the largest rectangle that can be inscribed in the right triangle... (answered by math_helper)

Find the area of the largest equilateral triangle that can be inscribed in a circle whose (answered by ikleyn)

Find the area of the largest right triangle that can be inscribed in a circle of radius... (answered by ikleyn)

FInd the area of the largest rectangle that can be inscribed in a semicircle of fadius... (answered by Edwin McCravy)

Find the volume of the largest cylinder that can be inscribed in a right circular cone of (answered by solver91311)

A circle can be inscribed in an equilateral triangle, each of whose sides has length... (answered by math_helper,ikleyn)

determine dimensions of the rectangle of a greatest area that can be inscribed in a... (answered by TimothyLamb)

Find the area of the largest rectangle with sides parallel to the coordinate axes which... (answered by Fombitz)