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

6. split the quadrilateral into two triangles
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=
K= m²
Measure of angle A

8. Use the law of cosines to find the length of c, the third side.
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= 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=
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°.
=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=
K= m²
87.7 °
72 m
196.3 m 6

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

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