# Smallest Triangular Portion From A Square Lot

**Problem**

A farmer owned a square field measuring exactly 2261 m on each side. 1898 m from one corner and 1009 m from an adjacent corner stands Narra tree. A neighbor offered to purchase a triangular portion of the field stipulating that a fence should be erected in a straight line from one side of the field to an adjacent side so that the Narra tree was part of the fence. The farmer accepted the offer but made sure that the triangular portion was a minimum area. What was the area of the field the neighbor received and how long was the fence? Hint: Use the Cosine Law.

A. A = 972,325 m^{2} and L = 2,236 m |

B. A = 950,160 m^{2} and L = 2,122 m |

C. A = 946,350 m^{2} and L = 2,495 m |

D. A = 939,120 m^{2} and L = 2,018 m |

**Answer Key**

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**Solution**

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**Largest rectangle inscribed in a given triangle:**

*BDNE*inscribed in the triangle

*ABC*

*BE*= 0.5

*BC*and

*BD*= 0.5

*BA*(

*see figure below*)

For mathematical proof, see Problem 32 of Maxima and Minima.

The same proportion is true if the rectangle is given and the triangle is unknown, which is the case of this problem.

**Smallest triangle that can circumscribe a given rectangle:**

*ABC*is circumscribing the rectangle

*BDNE*, for

*ABC*to be smallest

*BE*= 0.5

*BC*and

*BD*= 0.5

*BA*

It will follow that for minimum area of the triangle *ABC*, the Narra tree must be at the midpoint of the dividing fence *AC*.

By Pythagorean Theorem

*CEN*

$\left( L/2 \right)^2 = x^2 + y^2$

From triangle *BEN*

$\left( L/2 \right)^2 = 1009^2$

$L = 2018 ~ \text{m}$ ← Answer = [ D ]

For the area:

*BFN*

$1009^2 = 1898^2 + 2261^2 - 2(1898)(2261) \cos \theta$

$\cos = \dfrac{1898^2 + 2261^2 - 1009^2}{2(1898)(2261)}$

$\theta = 26.27^\circ$

From triangle *DFN*

$x = 1898 \sin \theta$

$x = 1898 \sin 26.27^\circ$

$x = 840 ~ \text{m}$

$2261 - y = 1898 \cos \theta$

$y = 2261 - 1898 \cos 26.27^\circ$

$y = 559 ~ \text{m}$

Area of triangle *ABC*

$A = 4\left( \frac{1}{2}xy \right) = 4 \times \frac{1}{2}(840)(559)$

$A = 939,120 ~ \text{m}^2$ ← *Check!*

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