# Plane Geometry

## 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 |

## Radius of Circle of New Atom Smasher

**Problem**

A new kind of atom smasher is to be composed of two tangents and a circular arc which is concave toward the point of intersection of the two tangents. Each tangent and the arc of the circle is 1 mile long, what is the radius of the circle? Use 1 mile = 5280 ft.

A. 1437 ft. | C. 1347 ft. |

B. 1734 ft. | D. 1374 ft. |

## Regular Octagon Made By Cutting Equal Triangles Out From The Corners Of A Square

**Problem**

A regular octagon is made by cutting equal isosceles right triangles out from the corners of a square of sides 16 cm. What is the length of the sides of the octagon?

A. 6.627 cm | C. 6.762 cm |

B. 6.267 cm | D. 6.276 cm |

## Smallest Part From The Circle That Was Divided Into Four Parts By Perpendicular Chords

**Problem**

Divide the circle of radius 13 cm into four parts by two perpendicular chords, both 5 cm from the center. What is the area of the smallest part.

## Area Bounded by Intersecting Chords in a Circle

**Problem**

Chords *AB* and *CD* intersect each other at *E* inside the circle. *AE* = 8 cm, *CE* = 12 cm, and *DE* = 20 cm. If *AB* is the diameter of the circle, compute the area of *AEC*.

A. 61.04 cm^{2} |
C. 39.84 cm^{2} |

B. 52.05 cm^{2} |
D. 48.62 cm^{2} |