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Algebra III Academic
Trigonometry Applications

Find the area of various triangles
Area formulas:
1
Right Triangle : A = bh
2
1
SAS Case : A = bcsin A
2
SSS Case : A = s(s - a)(s - b)(s - c), where s =
Given the Vertices of a Triangle: A =
(
1
a+b+c
2
)
1 x1 y1
1
1 x2 y2
2
1 x3 y3
Examples
Draw the triangle and find the area for the triangles that follow.
1. A triangles with vertices: (5, 7), (3, 2), (11, -3)
2. ABC a = 7.3, b = 10, c = 8.2
3. An isosceles triangle with leg lengths of 10 cm and base angles of 40°.
Algebra III Academic
Trigonometry Applications and Graphing
Class work
1. Find the area of a triangular flower garden
with sides 11 ft, 19ft, and 18ft, to the nearest
ten square feet.
4. The feds just launched two satellites and
Ivan Advantage to observe the satellites. As
Ivan approaches the two satellites, he finds
that one of them is 8 kilometers from him
and the other is 11 kilometers, and the angle
between the two (with Ivan at the vertex) is
120º. How far apart are the satellites?
2. A triangle has sides of lengths 12 in and 15
in, and the measure of the angle between
them is 24º. Find the area of the triangle.
3. To the nearest square foot, what is the area
of a triangular parcel of land, if two adjacent
sides measure 184 ft and 246 ft and the
angle between them measures 76º?
5. Three ships are assigned to rescue Ivan as
his spacecraft plunges into the ocean. The
ships are at the vertices of a triangle with
sides 5, 7, and 10 kilometers. Find the area
of ocean in the triangular region bounded by
the three ships. Find the largest angle of this
triangle.
6. Find the area of a triangular patio that has
sides 18ft, 20ft, and 22 ft, to the nearest ten
square feet.
5. Several orbits later as Ivan is about to re-enter, only Satellite No 1 is visible to Ivan, the other one being near
the opposite side of the Earth. He determines that Angle A measures 37º, Angle B measures 113º and the
distance between him and Satellite No 1 is 4362 kilometers. How far apart are Ivan and Satellite 2?
Ivan
world
Algebra III Academic
Trigonometry Applications and Graphing
Homework
Answer the following questions.
1. What is the area, to the nearest 0.1 acre, of a triangular field that is 529 ft on one side and 849 ft on
another, if the angle between these sides measures 102.7º?(1 acre =43,560 square feet)
2. A triangle has vertices: (5,7), (3,2), (8, -4) Find its area.
3. Solve this triangle. AFC : a  7, f  9, c  12. (You should use the Law of Cosines twice)
4. A triangle has vertices:  2, 3 ,  4,1 ,  3,7  . Find its area.
5. Determine the area of the shaded region.
4 cm
3 cm
Diameter
12 cm
2 cm
6. To the nearest square foot, what is the area of a triangular parcel of land, if two adjacent sides measure
184 ft and 246 ft and the angle between them measures 76º?
7. A surveyor measures the three sides of a triangular field and gets 114, 165, and 257 meters. What is the
measure of the largest angle of the triangle? What is the area of the field?
8. In DABC,a = 7.3,b =10.1,c = 8.2. Find its area.