The golden rectangle is inscribed in triangle ABC so that the longer side of a rectangle lies on the side AC. If AB = 7, BC = 11, and AC = 14, then what's the area of a rectangle?
A Golden Rectangle is one where the sides are in proportion of 1 : Phi
Where Phi = (1 + sqrt 5 ) / 2 ≈ 1.618
Find the area of the ltriangle
Semiperimeter, S = (7 + 11 + 14) /2 = 16
Area of triangle= sqrt [16 8 (16 - 70 (16 - 11) (16 - 14) ] = sqrt [ 16 * 9 * 5 * 2 ] = 12 sqrt (10)
We can find the height of the triangle thusly
Area = (1/2) AC * height
12sqrt (10) = 7 * height
(12/7) sqrt 10 = height ....call this BD
Let the rectangle be MNOP where AP + PO + OC = 14
Let the height of the rectangle = a and let PO = a * Phi
We can find DC as sqrt [ 11^2 - (12/7 * sqrt 10)^2 ] = 67/7
And we can find DA as sqrt [ 7^2 - (12/7 * sqrt 10)^2 ] = 31 / 7
By similar triangles
OC / a = DC / BD
OC = a * ( 67/7)/ [ (12sqrt 10 / 7 ] = 67 a / [ 12 sqrt 10]
AP / a = DA / BD
AP = 31a /[ 12sqrt 10 ]
So
AP + PO + OC =14
31 a / [ 12sqrt10 ] + (1 + sqrt 5] / 2 * a + 67a / [12sqrt 10 ] = 14
Solving for a we get that a ≈ 3.3329
So the area of the rectangle a * a * Phi = (3.3329)^2 * 1.618 ≈ 17.97 units ^2
A Golden Rectangle is one where the sides are in proportion of 1 : Phi
Where Phi = (1 + sqrt 5 ) / 2 ≈ 1.618
Find the area of the ltriangle
Semiperimeter, S = (7 + 11 + 14) /2 = 16
Area of triangle= sqrt [16 8 (16 - 70 (16 - 11) (16 - 14) ] = sqrt [ 16 * 9 * 5 * 2 ] = 12 sqrt (10)
We can find the height of the triangle thusly
Area = (1/2) AC * height
12sqrt (10) = 7 * height
(12/7) sqrt 10 = height ....call this BD
Let the rectangle be MNOP where AP + PO + OC = 14
Let the height of the rectangle = a and let PO = a * Phi
We can find DC as sqrt [ 11^2 - (12/7 * sqrt 10)^2 ] = 67/7
And we can find DA as sqrt [ 7^2 - (12/7 * sqrt 10)^2 ] = 31 / 7
By similar triangles
OC / a = DC / BD
OC = a * ( 67/7)/ [ (12sqrt 10 / 7 ] = 67 a / [ 12 sqrt 10]
AP / a = DA / BD
AP = 31a /[ 12sqrt 10 ]
So
AP + PO + OC =14
31 a / [ 12sqrt10 ] + (1 + sqrt 5] / 2 * a + 67a / [12sqrt 10 ] = 14
Solving for a we get that a ≈ 3.3329
So the area of the rectangle a * a * Phi = (3.3329)^2 * 1.618 ≈ 17.97 units ^2
The golden rectangle is inscribed in triangle ABC so that the longer side of a rectangle lies on the side AC. If AB = 7, BC = 11, and AC = 14, then what's the area of a rectangle?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Using the law of cosines, we find that ∠A ≈ 50.754º and ∠C ≈ 29.526º.
To calculate an x, we can use this equation:
(x / tanA) + (x * Φ) + (x / tanC) = 14 Φ = 1/2(1 + √5)
x ≈ 3.333
Rectangle area = x2Φ ≈ 17.974