9.3.3C Pythagorean Theorem & Right Triangles
Apply the Pythagorean Theorem and its converse to solve problems and logically justify results.
For example: When building a wooden frame that is supposed to have a square corner, ensure that the corner is square by measuring lengths near the corner and applying the Pythagorean Theorem.
Know and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems and logically justify results.
For example: Use 30-60-90 triangles to analyze geometric figures involving equilateral triangles and hexagons.
Another example: Determine exact values of the trigonometric ratios in these special triangles using relationships among the side lengths.
Overview
Standard 9.3.3 Essential Understandings
The study of geometric figures might be considered one of the "main topics" in geometry, along with, among others, proportional thinking, reasoning and sense making. The focus of this standard is on geometric figures, with the others listed above also being involved to a great extent.
In this standard, students will work with a transversal that intersects two lines. They will study angles formed by the three lines, and determine whether or not the two intersected lines are parallel. They will see many types of triangles, including "special" triangles such as isosceles, equilateral, right, 30-60-90 and 45-45-90 triangles, focusing on the properties of each of these figures.
Students will study similarity and understand that congruence is a special case of similarity. They will work with scale factors in "real-life" applications of similar figures. They will look at quadrilaterals and put them into a hierarchy, with "special" quadrilaterals being "nested" within others. Also, in this standard, students will study circles and their properties, with special focus on their applications.
Benchmark Group C - Pythagorean Theorem and Right Triangles
9.3.3
9.3.3.4 Apply the Pythagorean Theorem and its converse to solve problems and logically justify results.
9.3.3.5 Know and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems and logically justify results.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- Apply the Pythagorean Theorem to calculate the length of a side of a right triangle, given the other two side lengths;
- Apply the converse of the Pythagorean Theorem to determine whether a triangle is a right triangle;
- Calculate the missing side lengths of 30-60-90 triangles and 45-45-90 triangles, given one length.
Work from previous grades that supports this new learning includes:
- Differentiate between acute triangles, right triangles and obtuse triangles, given markings on the sides and/or angles of a triangle;
- Be familiar with "simplified radical form," especially as it relates to lengths of sides of (right) triangles.
NCTM Standards
Geometry
- Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships:
- Analyze properties and determine attributes of two- and three-dimensional objects;
- Explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them;
- Establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others;
- Use trigonometric relationships to determine lengths and angle measures.
- Use visualization, spatial reasoning, and geometric modeling to solve problems:
- Use geometric models to gain insights into, and answer questions in, other areas of mathematics;
- Use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture.
Common Core State Standards (CCSM)
- HS.G-CO (Congruence) Prove geometric theorems.
- HS.G-CO.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
- HS.G-CO.10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
- HS.G-CO.11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals
- HS.G-SRT (Similarity, Right Triangles, & Trigonometry) Define trigonometric ratios and solve problems involving right triangles.
- HS.G-SRT.7. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
- HS.G-SRT (Similarity, Right Triangles, & Trigonometry) Apply trigonometry to general triangles.
- HS.G-SRT.10. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
- HS.G-C (Circles) Understand and apply theorems about circles.
- HS.G-C.1. Prove that all circles are similar.
- HS.G-C.2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
- HS.G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
- HS.F-TF (Trigonometric Functions) Extend the domain of trigonometric functions using the unit circle.
- HS.F-TF.3. Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6.
Misconceptions
- Once students are introduced to 30-60-90 triangles and 45-45-90 triangles, they often assume that the relationships between side lengths hold for any right triangle.
- Once students are introduced to the Pythagorean Theorem, they sometimes assume the relationships between the side lengths hold for any triangle.
Vignette
In the Classroom
In this vignette, students explore the relationships between sides of a triangle and learn about the Pythagorean Theorem.
Teacher: You might remember from your middle school years learning about the Pythagorean Theorem. Then again, you might not remember. We're going to take a couple of minutes to show this theorem using our interactive geometry software, and then talk about some ways the theorem might be useful.
When you're using the Pythagorean Theorem, what sorts of figures are you working with?
Student 3: Triangles.
Teacher: Would you be using any type of triangle?
Student 3: No, just right triangles.
Teacher: Right. Like this:
Teacher puts the following on the screen:
Teacher: We're going to measure the lengths of the sides of this triangle.
Teacher: It might not appear that there is a relationship between these numbers, but let's look at it a different way. We're going to square the lengths of the legs, and add those squares together, and compare that number to the square of the length of the hypotenuse.
Teacher: As I move a vertex or a side of the triangle, what are you noticing?
Student 4: When you take one side length and square it, and add that to another side length, squared, it's the same as the third side length, squared.
Teacher: Would that be true no matter which side lengths, squared, you added together?
Student 7: No, I think you have to take the two shorter lengths, squared, and add them together, and that sum is equal to the longest side length, squared.
Teacher: Right, and in a right triangle, what do we call those lengths?
Student 7: The two shorter sides are legs, and the longest side is the hypotenuse.
Teacher: Great. So, can you tell the Pythagorean Theorem in your own words?
Student 7: I think so. If we're working with a right triangle, then
(length of one leg)2 + (length of the other leg)2 = (length of hypotenuse)2
Teacher: Excellent. The converse of this is true, too, so that if you have a triangle in which the side lengths are in that relationship, then you know that triangle is a right triangle.
The next day in class:
Teacher: We're now going to look at an equilateral triangle in a slightly different way. What do you know about an equilateral triangle?
Student 2: All the sides are congruent.
Teacher: Yes, and is there anything else special about it?
Student 2: All the angles are congruent. They each have a measure of 60 degrees.
Teacher: How do you know that?
Student 2: I know the sum of the measures of the angles is 180, and if all the angles are congruent, then I just divide 180 by 3 to get 60.
Teacher: Good. Here's an equilateral triangle, just like the one on your paper.
Now, if we were to construct the altitude from point A, what would that look like?
Student 5: It would be straight down from A, so that segment AM forms a right angle with segment CT.
Teacher: Right. Is there anything else special about that segment?
Student 7: Didn't we find last week that, since we're working with an equilateral triangle, the same segment is an altitude, an angle bisector and a perpendicular bisector?
Teacher: You have a good memory. When you drop that altitude from point A, you're forming two smaller triangles, triangle CAM and triangle TAM. What can you tell me about the two smaller triangles?
Student 5: They're congruent, because of Hypotenuse-Leg.
Teacher: Yes, they are. Because of that, what must be true about angle CAM and angle TAM? How about segment CM and segment MT?
Student 3: They're congruent angles, and congruent segments, because corresponding parts of congruent triangles are congruent.
Teacher: OK, a follow-up question for you: What is the measure of each of those angles?
Student 3: They are each 30-degree angles, because angle CAT is a 60-degree angle.
Teacher: Right. Each of the smaller triangles, triangle CAM and triangle TAM, is what is known as a 30-60-90 triangle, due to their angle measures. I'm going to focus on just triangle CAM, which looks like this:
Teacher: In your groups of three, each of you should pick a side length that you'd like for segment CA. For this activity, it might be easiest at first if you pick an even number for that length. Be sure that your length is different from the lengths your partners choose.
Once you've decided on a length for segment CA, you should be able to pretty easily determine the length of segment CM. How?
Student 1: We know that segment CM is half the length of segment CA.
Teacher: Right. Now, each of you should calculate the length of segment AM. How would you go about finding that length?
Student 5: If we know two side lengths of a right triangle, we could use the Pythagorean Theorem to find the third side.
Teacher: Great. Go ahead and do it. Don't round off that third length - keep it in simplest radical form.
The teacher walks around the room, observing and helping where appropriate. After a few minutes, the teacher calls the students back together.
Teacher: OK, tell me the lengths of the sides of your particular triangles. Start with the length of the hypotenuse, then give the length of the short leg, then the length of the long leg.
Student 6: My hypotenuse was 6 inches, the short leg was 3 inches and the long leg was $3\sqrt{3}$ inches.
Student 8: My hypotenuse was 10 cm, the short leg was 5 cm and the long leg was $5\sqrt{3}$ cm.
Student 2: My hypotenuse was 8 furlongs, the short leg was 4 furlongs and the long leg was $4\sqrt{3}$ furlongs.
Student 7: Will it always be like that?
Teacher: What do you mean?
Student 7: Will the hypotenuse always be twice as long as the short leg, and the long leg be $\sqrt{3}$ times as long as the short leg?
Teacher: It's great that you were able to find that pattern. Yes, that will always be true if you are working with a 30-60-90 triangle.
Teacher: Now, I have another question for you. We've talked about a "short leg" and a "long leg." How can you tell which is which? In the diagrams we had seen earlier, it had looked pretty obvious which was the long leg and which was the short leg, but how about if the triangle isn't drawn in an obvious manner, so that it looks like this instead?
Student 1: The shortest side of a triangle is opposite the smallest angle, and the longest side is opposite the largest angle. Since angle D is the 30-degree angle, it is the smallest in this triangle. Segment OG is opposite that angle, so we know that segment OG is the shortest side. Since angle G is the right angle, which means that segment DO is the hypotenuse, so segment DG must be the long leg.
Resources
Teacher Notes
- Be sure to emphasize that not all triangles have a Pythagorean relationship. It might be good to explore what inequality would make a triangle an acute triangle and what inequality would make a triangle an obtuse triangle. (If a, b and c are side lengths of a triangle, with c being the length of the longest side, and a2 + b2 = c2, then the triangle is a right triangle. If a2 + b2 < c2, then the triangle is an obtuse triangle, and if a2 + b2 > c2, then the triangle is an acute triangle.)
- Emphasize that the relationships between side lengths of a 30-60-90 triangle or a 45-45-90 triangle apply for only those types of triangles.
Lesson on the Pythagorean Theorem and its converse
Corner to Corner
In the first lesson of this unit, students use pattern recognition to determine that the length of a diagonal of a square is equal to the side length times $\sqrt{2}$. They then attempt to discover the Pythagorean theorem by examining similar patterns for rectangles. In Lesson 1, students explore the relationship between the lengths of the sides and diagonals of a square. Students will use their discoveries to predict the diagonal length of any square. In Lesson 2, students further explore square roots using the diagonals of rectangles. Using measurement, students will discover a method for finding the diagonal of any rectangle when the length and width are known, which leads to the Pythagorean Theorem.
Additional Instructional Resources
National Council of Teachers of Mathematics (NCTM). (2010).Focus in High School Mathematics: Reasoning and Sense Making in Geometry. Reston, VA: National Council of Teachers of Mathematics.
30-60-90 triangle: a right triangle whose acute angles measure 30 and 60.
45-45-90 triangle: a right triangle whose acute angles each measure 45.
altitude: a segment from a vertex of a triangle that is perpendicular to the line that contains the opposite side.
Reflection - Critical Questions regarding the teaching and learning of these benchmarks
- What other instructional strategies can be used to engage students with properties of triangles?
- How can manipulatives be used to help students visualize these abstract concepts?
- How can the instruction be scaffolded for students?
- What additional scaffolding should be provided to ELL students?
- Do the tasks that have been designed connect to underlying concepts or focus on memorization?
- How can it be determined if students have reached this learning goal?
- How can the lesson be differentiated?
Assessment
Teacher: Let's say a ladder is "safe" if it's placed at an angle of between 65 degrees and 80 degrees with the ground. Why would we say that this range of angle measures is "safe?"
Student 9: If it were placed at an angle of less than 65 degrees with the ground, then there would be a danger of the ladder sliding out from the bottom, slipping down along the wall and the person on it crashing to the ground.
Teacher: Right. What might happen if it were placed at an angle of greater than 80 degrees?
Student 9: Then the danger would be that the person on the ladder would fall over backward.
Teacher: Also right. Now, let's figure out how high on a wall a 25-foot-tall ladder would reach.
Note to teachers: This first scenario assumes the students have no knowledge yet of trigonometric ratios, and therefore must be given the distance from the ladder to the wall.
What might you do to start this, in order to solve the problem?
Student 5: I think I'd draw a sketch of the situation. It would look like this:
Teacher: OK, now how could you calculate h?
Student 7: We need more information.
Teacher: What else do you need?
Student 7: We need to know how far the ladder is from the wall.
Teacher: To the nearest tenth of a foot, that distance is 10.6 feet for the 65-degree angle, and it's 4.3 feet for the 80-degree angle. The figures look like this:
Student 2: So we can just use the Pythagorean Theorem to calculate h, right?
Teacher: Right.
Student 2: OK, so then, for the 65-degree angle, (10.6)2 + h2 = 252, and I think h is about 22.6 feet.
Teacher: You're right, to the nearest tenth of a foot. How about for the 80-degree angle?
Student 8: I used (4.3)2 + h2 = 252, and then h is about 24.6 feet.
Note to teachers: This second scenario assumes students can apply basic trigonometric ratios to this example. Again, the diagrams look like this:
What might you do to start this, in order to solve the problem?
Student 4: Could we use trigonometry?
Teacher: Sure. How would you use trigonometry?
Student 4: Well, if we know that we're working with angle T, the ladder is the hypotenuse and the distance up the wall is the opposite leg.
Teacher: How does that help?
Student 4: We know we'll have to use the sine function.
Teacher: OK. Apply it in this case.
Student 9: Since the sine ratio is $\frac{length\ of\ the\ opposite\ leg}{length\ of\ the\ hypotenuse}$, our equation would look like this:
$\sin 65^{\circ}=\frac{h}{25}$, and $h=25\sin 65^{\circ}$. That would make h approximately 22.7 feet, so that's the height the ladder would reach up the side of the building.
Student 1: And for the 80-degree angle, we would have sin $\sin 80^{\circ}=\frac{h}{25}$, and $h=25\sin 80^{\circ}$. That means h is approximately 24.6 feet, and that's how high the ladder would reach.
Differentiation
Keep a chart of the trigonometric ratios handy, or just the familiar "SOH CAH TOA" phrase.
Keep a list of vocabulary terms (and their definitions, along with diagrams) handy, for each student.
Extend the ladder activity in the Assessment to include inverse trigonometric relationships. For example, if you have a 30-foot ladder and need it to reach 28 feet on a wall, will the angle with the ground be within the "safe" range?
Parents/Admin
Students are: (descriptive list) | Teachers are: (descriptive list) |
analyzing equilateral triangles, in particular what happens when an altitude is drawn from a vertex. | leading discussion on equilateral triangles. |
calculating side lengths for a 30-60-90 triangle. | observing student work and helping when appropriate. |
applying the Pythagorean Theorem to real-life situations. |
|
Parent Resources
- Sophia geometry lessons
This social learning community website includes lessons on multiple topics. Most of these lessons were developed by teachers and reviewed.
- Khan Academy geometry lessons
This website offers a library of videos and practice exercises on multiple topics.
- Teacher Tube, YouTube
These websites include multiple uploaded lessons on most school topics.
- Mathematics textbooks
Many textbook publishers have websites with additional resources and tutorials. Check your child's textbook for a weblink.