Download Common Core Geometry Unit 2 Items to Support Formative

Geometry Items to Support Formative Assessment
Unit 2: Triangles, Proof, and Similarity
Part II: Right Triangle Trigonometry
Define trigonometric ratios and solve problems involving right triangles.
G.SRT.B.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.
Task
Land surveyors use a tool called a transit to accurately determine distances and angle measures between
selected points located on the ground. They are an invaluable resource to contractors, builders, and
community planners.
The image below was taken from the notes of a local land surveyor. She was hired to determine which
plot of land would be the best choice for building a new community road. The community planners
indicated that the vertical rise of the road could not exceed 60 feet due to local regulations.
Review each of the plots surveyed and make a recommendation about which piece(s) of land would serve
as viable options for the road. Use mathematics to defend your recommendation.
Plot 1:
Plot 2:
Plot 3:
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Item 1
Two right triangles with different side lengths each have angles measuring 45°, 45°, and 90°. What might
the side lengths of each of these triangles be?
ITEM 2
How tall is a tree that cast a 30m shadow that makes a 53° angle with the ground?
Solution:
Tanθ = x/30
30tanθ = x
x =39.811m
ITEM 3
Portia claims that, for any 30-60-90 right triangle, the
Examine the given pair of 30-60-90
right triangles. Do the figures support Portia’s claim? Why or why not?
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Define trigonometric ratios and solve problems involving right triangles.
G.SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles.
Task
GIVEN:
is a right triangle
Prove that the
.
Answer:
1. Angle C = 90 degrees
2. A+B+C=180°
3. A+B+90°=180°
4. A+B=90°
5. A and B are complementary
6.
1.
2.
3.
4.
5.
6.
7.
7. Definition of Cosine
Definition of a right angle
Triangle Sum Theorem
Substitution
Subtraction Property of Equality
Definition of Complementary Angles
Definition of Sine
8.
8. Substitution
Item 1
Place a checkmark next to each value that is equivalent to
Item 2
Draw and label a right triangle to justify why the
and symbols in your justification.
is equivalent to
. Use words, numbers,
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under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Answer: Students may create various responses but each response should illustrate that the “opposite”
side for a given angle is the same as the adjacent angle for its complementary angle. Therefore, the
“opposite” side referenced in
is the same side referenced as “adjacent” for
.
Item 3
For each sample, write an equivalent trigonometric expression.
____________________
____________________
____________________
____________________
____________________
____________________
Answers:
Item 4
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Create a graphic organizer to group all of the equivalent expressions below. (NOTE: Some expressions
may belong to more that one group)
Answers
Group 1: sin 60°, cos 30°, √3/2, .8660
Group 2: sin30°, cos60°, ½, .5
Group 3: cos15°, sin75°, .9659
Group 4: cos75°, sin15°, .2588
G.SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangle in applied
problems.
Task
An airplane flying at 26,400 feet and heading for BWI has been commanded to land at the nearest airport
due to a major weather event in Baltimore. The pilot takes two readings to determine which airport is
closest at the time of the announcement. Pittsburgh International Airport is located to the north with an
angle of elevation reading at . Dulles International Airport is located to the southeast with an angle of
elevation reading at
. Which airport is closest to the plane?
Item 1
How might you use trigonometric ratios to calculate the area of the given regular polygon?
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Possible Solution: Draw in the segment from the center to the corner creating a 30°-60°-90°
triangle, with base of 4 cm (half the side of the hexagon.) They can use tan60°= x/4 or
tan30°=4/x to solve for the height of the triangle. (x = 6.928cm) Then they can find the area of
the triangle and multiply by 12. (1/2*4*6.928*12=166.277)
Item 2
How might you use trigonometric ratios to calculate the area of the given regular polygon?
Possible Solution: If you draw in the segment from the center to the corner adjacent to the given segment.
That will create an isosceles right triangle that is 45°-45°-90°. From there they can either use special
right triangles, pythagorean theorem, sin45° or cos45° to find the length of the side of the square and then
square that value. s = 12, A = 144
Item 3
In the late 1900s, engineers would utilize trigonometry to determine the distance across a mountain
canyon. Engineers would take a sighting from a point on one side of the canyon to a landmark on the
other side. They would then measure the distance from their original position to a point across from the
sighted landmark. Use the graphic below and given measurements to determine the length of bridge that
is needed to cross the canyon. (NOTE: AB = 27 miles)
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Source: Googlemaps
Item 4
Use trigonometric ratios to estimate the height of the building.
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under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.