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Discrepancy in Property Survey: Area Calculation of Quadrilateral Lot

In a building permit application, my parents needed a property survey for their quadrilateral lot. The surveyors calculated the area as 2.5 acres based on measurements and angles, but my parents disagreed, believing the area was larger. I undertook the task of recalculating the area by dividing the lot into two triangles and applying the appropriate mathematical formulas. After thorough calculations, the accurate area was determined to be 3.2 acres, revealing a significant discrepancy from the surveyors' original figure.

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Discrepancy in Property Survey: Area Calculation of Quadrilateral Lot

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  1. Surveyors Jordan Tully

  2. the problem

  3. In order to apply for a building permit, my parents had to get their property surveyed. They live on a quadrilateral shaped lot, as shown in the diagram below. 86m 113° 131° 146m 72m

  4. The surveyors took the measurements of the three sides and two included angles. From this information they were able to determine the area of the yard. The surveyors said that the area was 2.5 acres. However, my parents didn’t agree. They thought they had more land than that, so they asked me to determine the area of the yard.

  5. the question Did the surveyor calculate the area of the yard correctly?

  6. step 1 split the quadrilateral into two triangles 86 m 113° 146 m 72 m

  7. Find the area of the first triangle using the formula K= ½ b c sinA 2 Length of side b Length of side c K= ½ (146)(86)sin(113) K=5778.92947 K=5778.9 m² Measure of angle A

  8. 86m Use the law of cosines to find the length of c, the third side. 113° 146m c 3 c²= a² + b² - 2ab cosC c²=146² + 86² - 2(146)(86)cos(113) c=196.275419 m c=196.3 m

  9. 113° A step 4 146 m 196.3 m a ÷ sinA = b ÷ sinB 196.3 ÷ sin(113)=146 ÷ sinB B=43.20685 B=43.2° Since you have the length of the third side, use the law of sines to find the measure of angle A.

  10. 131° #5 43.2° B The measure of the whole angle before the split was 131°. To find the measure of angle B subtract 43.2° from 131°. 131 - 43.2=87.8°

  11. Using angle B, the length of side c, and the given side of 72 m; find the area of the second triangle. K=½ b c sinA K= ½(196.3)(72)sin(87.8) K=7061.591178 K=7061.6 m² 87.7 ° 72 m 196.3 m 6

  12. Add the two areas together to get a total area. 5778.9 m² + 7061.6 m² = 12846.5 m² 5778.9 m² step #7 7061.6 m²

  13. step 8 Because there are 4047 m² in an acre, divide the total area by 4047 m². 12846.5 ÷ 4047 = 3.174326662 = 3.2 acres 12846.5 m²

  14. the answer The surveyors went wrong somewhere in their calculations because as our solution proved, the land is a total of 3.2 acres not 2.5 acres as the surveyors had said. 3.2 ≠ 2.5

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