Question

An antique map was found in the attic of a local courthouse. It shows some measurements

from a local farm that was divided into 5 parts. Some of the measurements have faded with

age, so you must find the remaining measurements, as well as calculate the total area and

perimeter of the outside of the property. All lengths are measured in miles.

Answer 1

Perimeter: 26.2611 miles

Total area: 41 miles

Explanation:

The map can be seen in the figure attached.

From the Pythagorean Theorem:

AC^2 = 3^2 + 4^2

AC = sqrt(25)

AC = 5

From the Law of Cosines,

AD^2 = 3.1623^2 + 5^2 - 2*3.1623*5*cos(71.5651°)

AD = sqrt(25)

AD = 5

From the Law of Sines:

7.8102/sin(∠EDA) = 5/sin(38.8845°)

7.8102*sin(38.8845°)/5 = sin(∠EDA)

arcsin(0.9805) = ∠EDA

101.3117° = ∠EDA (the obtuse solution)

The addition of the 3 angles of a triangle must be equal to 180°, then:

∠EAD = 180° - 38.8845° - 101.3117° = 39.8038°

From the picture:

∠EFA + ∠GFA = 180°

∠GFA = 180° - 120.9638°

∠GFA = 59.0362°

The addition of the 3 angles of a triangle must be equal to 180°, then:

∠GFA + ∠FGA + ∠GAF = 180°

∠GAF = 180° - 59.0362° - 90°

∠GAF = 30.9638°

From the picture:

∠FAE + ∠GAF + ∠EAD = 90°

∠FAE = 90° - 30.9638° - 39.8038°

∠FAE = 19.2324°

From Law of Sines:

ED/sin(39.8038°) = 5/sin(38.8845°)

ED = 5*sin(39.8038°)/sin(38.8845°)

ED = 5.0988

From Law of Sines:

EF/sin(19.2324°) = 7.8102/sin(120.9638°)

EF = 7.8102*sin(19.2324°)/sin(120.9638°)

EF = 3

From definition of Sine:

sin(∠GAF) = GF/5.831

GF = sin(30.9638°)*5.831

GF = 3

From definition of Cosine:

cos(∠GAF) = GA/5.831

GA = cos(30.9638°)*5.831

GA = 5

Perimeter = 3+4+3.1623+5.0988+3+3+5=26.2611 miles

Area of the triangles:

ΔABC = (1/2)*3*4 = 6 miles

ΔADC = (1/2)*5*3.1623*sin(71.5651°) =7.5 miles

ΔADE = (1/2)*7.8102*5.0988*sin(38.8845°) = 12.5 miles

ΔAEF = (1/2)*3*5.831*sin(120.9638°) = 7.5 miles

ΔAFG = (1/2)*5*3 = 7.5 miles

Total area = 6 + 7.5 + 12.5 + 7.5 + 7.5 = 41 miles