Download G.8 - DPS ARE

Geometry: Similarity, Right Triangles, and
Trigonometry — G-SRT
ELG.MA.HS.G.7: Define trigonometric ratios and
solve problems involving right triangles.
 G-SRT.C.6 Understand that by similarity, side
ratios in right triangles are properties of the
angles in the triangle, leading to definitions of
trigonometric ratios for acute angles.
 G-SRT.C.7 Explain and use the relationship
between the sine and cosine of complementary
angles.
 G-SRT.C.8 Use trigonometric ratios and the
Pythagorean Theorem to solve right triangles in
applied problems.★
Geometry: Similarity, Right Triangles, and
Trigonometry — G-SRT
ELG.MA.HS.G.8: Apply trigonometry to general
triangles.
 G-SRT.D.9 (+) Derive the formula A = ½ ab sin (C)
for the area of a triangle by drawing an auxiliary
line from a vertex perpendicular to the opposite
side.
 G-SRT.D.10 (+) Prove the Laws of Sines and
Cosines and use them to solve problems.
 G-SRT.D.11 (+) Understand and apply the Law of
Sines and the Law of Cosines to find unknown
measurements in right and non-right triangles
(e.g., surveying problems, resultant forces.)
Students will demonstrate command of the ELG by:
1

Deriving the formula 𝐴 = 𝑎𝑏𝑠𝑖𝑛𝐶 for the area of a triangle.



Proving the Laws of Sines and Cosines.
Applying the Laws of Sines and Cosines to solve problems.
Applying the Laws of Sines and Cosines to find unknown measurements in triangles.
2
Vocabulary:



Auxiliary line
Law of Cosines
Law of Sines
Sample Assessment Questions:
1) Standard(s): G-SRT.D.11
Source: Key Data Systems
Item Prompt:
Three buoys, X, Y, and Z, are placed in the ocean to mark a sailboat race. Contestants start at X, sail to buoys Y and Z, and then return to buoy X. The location of the buoys
and the racecourse are shown below.
Part A:
What is the distance between buoy X and buoy Z to the nearest tenth of a mile?
Part B:
Explain how you determined your answer.
Correct Answer(s):
Part A:
3.1 miles
Part B:
𝑠𝑖𝑛 40°
From the Law of Sines,
=
2.4
2) Standard(s): G-SRT.D.11
𝑠𝑖𝑛 56°
𝑦
. Rearranging leads to y=
2.4 (𝑠𝑖𝑛56°)
sin 40°
≈ 3.1. Therefore, the distance between buoys X and Z is approximately 3.1 miles.
Source: Key Data Systems
Item Prompt:
A pasture is in the shape of a triangle. Two sides of this pasture measure 3.4 miles and 2.6 miles, and the angle between these two sides is 40°.
Part A:
To the nearest tenth of a mile, determine the length of the third side of the pasture.
Part B:
Explain how you determined your answer.
Correct Answer(s):
Part A:
2.2 miles
Part B:
The Law of Cosines states that c2 = a2 + b2 – 2ab · cos C, where C is the angle between side a and side b. Letting a = 3.4 and b = 2.6, substitution yields c2 = 3.42 + 2.62 –
2(3.4)(2.6)(cos 40). Simplifying leads to c2 = 11.56 + 6.76 – 17.68(cos 40) ≈ 4.776. Taking the square root yields c ≈ 2.2 miles.