Geometry Items to Support Formative Assessment
Unit 2: Triangles, Proof, and Similarity
Part II: Right Triangle Trigonometry
Apply trigonometry to general triangles. (Geometry GT)
G.SRT.D.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems.
Task
Prove that, for any triangle with angle measure A, B, and C and corresponding side lengths a, b, and c,
that
sin A
a
=
sin B
b
=
sin C
.
c
Answer: One possible proof can be interactively studied at
http://jwilson.coe.uga.edu/emt668/emat6680.2001/mealor/emat%206700/law%20of%20sines/Law%20of
%20Sines%20proof%201/lawofsinesproof1.html
Item 1
Solve for all missing sides and angle measures for the following triangle.
Answer: The measure of angle C is 80 degrees since the sum of the other two angles is 100 degrees and
the sum of all interior angles of a triangle sum to 180 degrees. In order to find the lengths of the missing
sides (b and c), we will apply the Law of Sines (AAS) as follows:
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Item 2
Solve for all missing sides and angle measures for the following triangle.
Answer: This problem requires the Law of Sines (SSA) for solution. Begin by solving for the measure of
angle B as follow….
Now that we know that the measure
of two angles (19.2 and 50 degrees) we can find the measure of
∘
the third angle. (180 - (69.2)) = 110.8 . So the measure of angle C is 110.8∘ .
We need to use the Law of Sines once more to find the measure of the third side.
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Item 3
Solve for all missing sides and angle measures for the following triangle.
Answer: This SAS case requires the Law of Cosines to solve.
First we will solve for the length of the third side as follows…
c ≈ 5.13
Now we can solve for one of the missing angle measures.
Using the first two angle measures (46.93 and 30 degrees) we can find the measure of the third angle. B =
180 - 76.93; B = 103.07∘
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Item 4
Solve for all missing sides and angle measures for the following triangle.
Answer: This SAS case requires the Law of Cosines to solve.
First we will solve for one of the missing angle measures.
Next, we will solve for a second missing angle measure.
The final angle can be found using the Triangle Sum Theorem.
C = 180 - (101.5+ 44.4) = 34.1 degrees
Apply trigonometry to general triangles. (Geometry GT)
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).
Task
Prove that, for any triangle with angle measure A, B, and C and corresponding side lengths a, b, and c,
that c 2 = a2 + b2 − 2ab cos C, b2 = a2 + c 2 − 2ac cos B, and a2 = b2 + c 2 − 2bc cos A respectively.
Answer: A proof of the Law of Cosines can be found at http://mathproofs.blogspot.com/2006/06/law-ofcosines.html
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under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Item 1
A shipping company wants to include aerial shipping between its three locations; Altoona, PA,
Washington, DC, and Roanoke, VA. Using the graphic below, calculate the missing side length (distance
from Roanoke to Altoona) and missing angle measures (for navigation purposes).
Source: Googlemaps
Item 2
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under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Air traffic control specialists rely on technology to gather information about incoming flights. For BWI,
the specialists receive data from two radar sensors place on the ground 2 miles from the airport. The
sensors are located on the ground and placed 1000 feet apart. For one incoming flight, sensor one
reported that the angle of elevation for the plan was 25∘ . At the same time, the second sensor reported an
angle of elevation at 30∘ . Determine the altitude of the plane at the time of the reading.
Answer: Students may represent the problem by creating a drawing such as the one below.
Item 3
Solve for the missing side length in the figure to calculate the approximate height of the St. Louis Arch.
(NOTE: Figure is not drawn to scale)
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Item 4
The roof of a lego doll house is created with two 5-inch straight pieces adjoined at a 60∘ angle. How far
apart must the builder erect the house walls to ensure a perfect roof-to-wall fit?
Answer: Students may represent the problem by creating a drawing such as the one below.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product
under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product
under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.