Geometry of Right Triangles
Geometry of Right Triangles
Objectives
In this unit, students will learn formulas to solve for sides and angles in right triangles. Students will:
- use the Pythagorean Theorem to solve for missing sides in right triangles.
- use the trigonometric ratios to solve for missing sides and angles in right triangles.
- learn how to find the slope of a line.
- use the Distance Formula and learn how it relates to the Pythagorean Theorem.
- learn how to find the midpoint of a line.
Essential Questions
- How can you explain the relationship between congruence and similarity in both two and three dimensions?
- How are coordinates manipulated algebraically to represent, interpret, and verify geometric relationships?
Related Unit and Lesson Plans
Related Materials & Resources
The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.
- Cloud pictures
http://www.pals.iastate.edu/carlson/main.html
- Show students the objective of tangrams or, if students have computer access, allow them to do a puzzle on the computers
https://pbskids.org/cyberchase/games/tangram-game
- A set of tangrams and the length of some of the sides
http://mathworld.wolfram.com/Tangram.html
- Demonstration of how the shapes are rotated and shifted to create a “cat”
http://demonstrations.wolfram.com/Tangram/
- Information regarding the slide’s angle with the ground should be 30° or less, and a teeter-totter should make a 25° angle
http://orthoinfo.aaos.org/topic.cfm?topic=A00333
- Slide picture
http://www.swingsetmall.com/
www.purplemath.com
- Image of the coordinate plane for the Graphic Organizer in Lesson 3
http://www.geometry.uconn.edu/5th%20grade%20geometry/TheCoordinatePlane.htm
- U.S. map for the Lesson 3 Exit Ticket
http://mapsof.net/united_states/static-maps/gif/map-of-the-united-states-of-america
Formative Assessment
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View
1. The Pythagorean Theorem only works for what type of triangle?
A
Acute
B
Equilateral
C
Obtuse
D
Right
2. The hypotenuse of a right triangle is 20, and one of the legs is 12. What is the length of the other leg?
A
8
B
16
C
256
D
544
3. Georgia goes to the middle school 4 miles from home, and Nathaniel goes to the elementary school 5 miles from home. Using the picture, determine how far apart the schools are from one another.
A
6.4 miles
B
9 miles
C
20.5 miles
D
41 miles
4. What is the name of the ratio that uses the opposite side and the hypotenuse?
A
Cosine
B
Sine
C
Tangent
D
Trigonometric
5. Solve for x.
A
0.04
B
17.98
C
19.10
D
22.25
6. A school building is 150 feet tall, and it casts a shadow 95 feet long. Use the picture to find the angle of the sun.
A
32.35°
B
39.30°
C
50.70°
D
57.65°
7. What is the distance between the points (7, 3) and (5, 4)?
A
B
5
C
D
95
8. What is the midpoint between the points (−2, 5) and (6, −3)?
A
(1, 2)
B
(2, 1)
C
(−4, 4)
D
(4, −4)
9. What is the slope of the line going through the points (−6, 8) and (−4, 5)?
Multiple-Choice Answer Key
1. D
2. B
3. A
4. B
5. D
6. D
7. A
8. B
9. C
Short-Answer Items:
10. Billy rode his bike 3 miles west and then 6 miles north to get from school to home. How far would he have traveled if he had taken the northwestern route? Leave your answer in simplified radical form.
11. An airplane travels 45,000 feet to get to its cruising altitude of 30,000 feet. Use the diagram to find the angle at which the plane travels to reach cruising altitude.
12. Using the points (2, 3) and (−3, −9), find the distance between them, the midpoint, and the slope of the line going through them.
Short-Answer Key and Scoring Rubrics:
10. Billy rode his bike 3 miles west and then 6 miles north to get from school to home. How far would he have traveled if he had taken the northwestern route? Leave your answer in simplified radical form. (3√5 miles)
Points
Description
3
The student used the Pythagorean Theorem correctly, left the answer in simplified radical form, and included the units.
2
The student used the Pythagorean Theorem correctly, included the units, but wrote the answer as a decimal.
1
The student used the Pythagorean Theorem correctly, but wrote the answer as a decimal, and did not include the units.
0
The student did not use the Pythagorean Theorem correctly and therefore got the incorrect answer.
11. An airplane travels 45,000 feet to get to its cruising altitude of 30,000 feet. Use the diagram to find the angle at which the plane travels to reach cruising altitude. (41.8°)
Points
Description
3
The student used the correct trigonometric ratio to solve for the angle and included the units in the answer.
2
The student used the correct trigonometric ratio to solve for the angle but did not include the units in the answer.
1
The student did not use the correct trigonometric ratio to solve for the angle but included the units in the answer.
0
The student did not use the correct trigonometric ratio and did not include the units.
12. Using the points (2, 3) and (−3, −9), find the distance between them, the midpoint, and the slope of the line going through them.
Points
Description
3
The student calculated the distance, midpoint, and slope correctly.
2
The student calculated two of the three correctly.
1
The student calculated one of the three correctly.
0
The student didn’t calculate any of them correctly.
Performance Assessment:
Use a blank sheet of graph paper to perform the following.
1. Plot the points A (−14, 3), B (10, −4), C (−14, −4), and connect the points.
2. Solve for the hypotenuse using the Distance Formula.
3. Solve for the hypotenuse using the Pythagorean Theorem.
4. Find the slope of the hypotenuse.
5. Find the midpoint of both legs of the triangle.
6. Solve for the angle at point A using the trigonometric functions sine, cosine, and tangent.
7. Solve for the angle at point B using the trigonometric functions sine, cosine, and tangent.
Performance Assessment Key and Scoring Rubric:
1.
2. 25
3. 25
4.
5. (−14, −½) and (−2, −4)
6. 73.7°
7. 16.3°
Points
Description
5
The student:
- plotted the points correctly and connected the points.
- solved for the hypotenuse correctly using both methods.
- found the slope correctly.
- solved for the midpoints correctly.
- solved for both angles correctly and labeled the degrees.
4
The student:
- plotted the points correctly and connected the points.
- incorrectly solved for one out of the four (hypotenuse, slope, midpoint, angles).
3
The student:
- plotted the points correctly and connected the points.
- incorrectly solved for two out of the four (hypotenuse, slope, midpoint, angles).
2
The student:
- plotted the points correctly and connected the points.
- incorrectly solved for three out of the four (hypotenuse, slope, midpoint, angles).
1
The student:
- plotted the points correctly and connected the points.
- incorrectly solved for four out of the four (hypotenuse, slope, midpoint, angles).
0
The student did not plot the points correctly and therefore did not correctly answer the other questions.
DRAFT 10/13/2011
