Download Module 2 Lesson 31 with Notes - Greeley

GEOMETRY
MODULE 2 LESSON 31
USING TRIGONOMETRY TO DETERMINE AREA
OPENING EXERCISE
Three triangles are presented below. Determine the areas for each triangle, if possible. If it is not
possible, describe what is need in order to determine the area.
1
(5)(12) = 30
2
1
𝐴𝑟𝑒𝑎 ∆𝐷𝐸𝐹 = (20)(8) = 80
2
𝐴𝑟𝑒𝑎 ∆𝐴𝐵𝐶 =
There is not enough information to find the height of ∆𝐺𝐻𝐼 and thus we cannot find its area.

Is there a way to find the missing information? No
DISCUSSION
Consider ∆𝐺𝐻𝐼 given a value for side GH. Could we find the area of ∆𝐺𝐻𝐼now?
MOD2 L31
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Find a value for x.
ℎ2 + 𝑥 2 = 72
ℎ2 + (15 − 𝑥)2 = 122
ℎ2 = 49 − 𝑥 2
ℎ2 = 144 − (15 − 𝑥)2
49 − 𝑥 2 = 144 − (15 − 𝑥)2
49 − 𝑥 2 = 144 − (225 − 30𝑥 + 𝑥 2 )
49 − 𝑥 2 = 144 − 225 + 30𝑥 − 𝑥 2
130 = 30𝑥
𝑥=
13
3
Substitute to find height.
13 2
ℎ = 49 − ( )
3
169
ℎ2 = 49 −
9
272
ℎ2 =
9
2
ℎ= √
272 4√17
=
9
3
Finally, we calculate area.
1
4√17
𝐴𝑟𝑒𝑎 = (15) (
) = 10√17
2
3
Let’s consider representing area for such triangles in general.
𝐴𝑟𝑒𝑎 =
1
𝑎ℎ
2
1
𝑏
𝐴𝑟𝑒𝑎 = 2 𝑎 ℎ (𝑏) Multiplication by 1
1
ℎ
𝑏
𝐴𝑟𝑒𝑎 = 2 𝑎 (1) (𝑏)
1
ℎ
𝐴𝑟𝑒𝑎 = 2 𝑎 𝑏 (𝑏) Terms are rearranged by the value is the same.
MOD2 L31
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
ℎ
With respect to 𝜃, what does 𝑏 represent?
ℎ
= sin 𝜃
𝑏

Make the substitution to the last line from above.
𝐴𝑟𝑒𝑎 =
1
𝑎𝑏 sin 𝜃
2
where a is the base and b is the “other” side to 𝜃

Which part of the expression above represents the height?
ℎ = 𝑏 sin 𝜃
PRACTICE
A farmer is planning how to divide his land for planting next year’s crops. A triangular plot of land is
left with two known side lengths measuring 500m and 1700 m. These sides create a 30° angle. Draw
and label the triangle then find the area.
1
𝐴𝑟𝑒𝑎 = 𝑎𝑏 sin 𝜃
2
1
𝐴𝑟𝑒𝑎 = (1700)(500) sin 30
2
𝐴𝑟𝑒𝑎 = 212,500 𝑠𝑞. 𝑚𝑒𝑡𝑒𝑟𝑠
WORKBOOK
Exercise 1: A real estate developer and her surveyor are searching for their next piece of land to build
on. They each examine a plot of land in the shape of ∆𝐴𝐵𝐶. The real estate developer measures the
length of AB and AC and finds them to both be approximately 4,000 feet, and the included angle has a
measure of approximately 50°. The surveyor measures the length of AC and BC and finds the lengths
to be approximately 4,000 feet and 3,400 feet, respectively, and measures
the angle between the two sides to be approximately 65°.

Draw a diagram that models the situation. Label all lengths and angle
measures.
MOD2 L31
3

Find the area of the triangle. Round your answer to the nearest whole number. NOTE: There are
two possible answers.
1
𝑎𝑏 sin 𝜃
2
1
𝐴𝑟𝑒𝑎 = (4000)(4000) sin 50
2
𝐴𝑟𝑒𝑎 =
𝐴𝑟𝑒𝑎 = 6,128,356 𝑠𝑞. 𝑓𝑒𝑒𝑡
1
𝑎𝑏 sin 𝜃
2
1
𝐴𝑟𝑒𝑎 = (3400)(4000) sin 65
2
𝐴𝑟𝑒𝑎 =
𝐴𝑟𝑒𝑎 = 6162893 𝑠𝑞. 𝑓𝑒𝑒𝑡
SUMMARY

To determine the area of a triangle when height is not provided use
𝟏
𝑨𝒓𝒆𝒂 = 𝟐 𝒂𝒃 𝐬𝐢𝐧 𝜽 , where a is the base and b is the “other” side to 𝜃.
NOTE: The angle used must be the angle created by the two known sides.
HOMEWORK
Problem Set Module 2 Lesson 31, page 234
#1 thru #4, #6, #8, and #10
Show all work in an organized and linear manner.
DUE: Tuesday, Jan 30, 2017
MOD2 L31
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