Math10th grade LEARNING OBJECT LEARNING UNIT Trigonometry, a study of Identifying properties of triangles the measurement of the angle through function S/K SCO: Understand the isosceles triangle theorem. SKILL 1. Identify triangles with sides of equal measurement. SKILL 2. Identify triangles with equal angles. SKILL 3. Distinguish the metric characteristics of isosceles right triangles. SKILL 4. Understand the hypothesis stated in the isosceles triangle theorem and its inverse. SKILL 5. Solve geometric situations by applying the isosceles right triangle theorem. SCO: Identify the 30-60-90 theorem. SKILL 6. Understand the relationship of constant ratio between the hypotenuse and a leg. SKILL 7. Identify the relationships of measurement between the angles and catheti of the triangle. SKILL 8. Illustrate 30-60-90 triangles by applying the relationship between the measurements between the catheti. SCO: Understand the Pythagorean Theorem. SKILL 9. Recognize the hypotenuse as the longest leg in a right triangle. SKILL 10. Interpret the Pythagorean Theorem in problem solutions. SKILL 11. Solve measurement situations involving right triangles. SKILL 12. Make an argument supporting the use of the procedure in problem solving. Language English Socio cultural context of 10th grade students will find situations in their the LO context related with the measurement of heights and lengths of sides of physical structures related to construction, such as pyramids, towers, ramps and others. Curricular axis Variational thinking and analytic and algebraic systems. Spatial thinking and geometric systems. Standard competencies Recognize and contrast properties and geometric relationships used to demonstrate basic theorems. Background Knowledge Interpretation of situations by means of the Pythagorean Theorem. Classification and categorization of triangles. Basic Learning Use the Pythagorean Theorem to verify if a Rights triangle is a right triangle or not, as well for problem solving. English Review topic Simple Present vs Present Perfect Glossary Triangle: A flat figure formed by three vertexes and three sides. Straight Triangle: A triangle that has an internal right angle. Isosceles Triangle: A triangle that is defined by having two sides of equal length. Catheti: The sides opposing the acute angles of the right triangle. Hypotenuse: The diagonal, or longest side, of the right triangle. Pythagorean Theorem: The theorem that states that any right triangle presents the relation 푐2 = 푎2 + 푏2, c being the hypotenuse and a and b being the catheti. Length: The distance between two points. Right angle: An angle that measures exactly 90°. Vocabulary Box Acute: (Geometry) a. (of an angle) less than 90°. b. (of a triangle) containing only acute angles. Congruent: (Geometry) Coinciding at all points when superimposed. Foreman: A person in charge of a particular department, group of workers, etc., as in a factory or something similar. Fulfill: To satisfy (requirements, obligations, etc.) Oblique: Neither perpendicular nor parallel to a given line or surface; slanting, sloping. Obtuse Angle: An angle greater than 90° but less than 180° (Terms retrieved from the website www.dictionary.com) NAME: _________________________________________________ GRADE: ________________________________________________ Printable Resource Introduction Triangles and their importance Very relevant knowledge has always been linked to triangles. From the study of these geometric shapes many aspects of engineering are derived, such as methods used to calculate distance or center of gravity, navigation systems, tools and and mechanisms for work, among others. Pyramids represent an example that allows us to distinguish these various elements. Figure. Pyramids. This image shows different aspects: first, all of the visible sides are isosceles triangles (remember that there is a theorem about their characteristics). The second aspect to a famous Greek mathematician: Thales of Miletus. He applied an ingenuous method to calculate the height of a pyramid with the help of trigonometry and the similarities between triangles, using their shadows and the height of a wooden stick as a means of Figure. Right triangles in a pyramid. comparing them. Additionally, elements of the Pythagorean Theorem can also be used. Pythagoras was another ancient mathematician and philosopher. The theorem named after him can be applied to right triangles such as the ones showed in the next image. Undoubtedly, this is proof of its importance since early times. Looking at the following structures, can you see the triangles? Why do you think they are so important? Triangles are present in the basic principles of geometry because every single polygon can be broken down into or formed by triangles. We will study some of the theorems regarding triangles and their applications. Objectives To recognize the properties that satisfy the requirements of some special right triangles to be considered as such. To identify the properties of isosceles triangles. To distinguish the 30-60-90 theorem and the relationships between the measurements of each side. To apply the Pythagorean Theorem for the task of finding measurements. Very important! The sum of the internal angles of any triangle is exactly 180°. Activity One Isosceles triangles Oscar and Miguel are friends from school and are having a conversation. Miguel: Hello Oscar, how are you? I have not seen you lately, why have you not come to school? Oscar: My parents and I were travelling, and we returned yesterday. What topics have you Figure. covered? Miguel: We have worked a lot on geometry. Our last class was about triangle theorems. Oscar: Is it a vast topic? Could you help me? Miguel: Of course, I will explain it to you. We should begin with the isosceles triangle theorem. First, we must know if you have any background knowledge. Answer this question: what is an isosceles triangle? Oscar: Oh, I know that! I will draw it and explain it to you. An isosceles triangle is one with two equal sides, called catheti or legs, like the one I just drew. Miguel: Yes, very good, but you are missing something. Oscar: Let me think… Errrr, I cannot think of anything! Miguel: When we talk about triangles, we say that the angles in the base are equal, or in other words, are congruent. Oscar: Oh yes! That is one of the characteristics. Miguel: This characteristic defines the isosceles triangle theorem, which is: “In an isosceles triangle, the angles opposing the equal sides are equal” . And is written as: ∡퐴퐵퐶 ≅ ∡퐴퐶퐵 Oscar: Ok, that is easy, but what is its function? Miguel: Look at an example: Find the value of the missing angles in the following figure: Given that the angle ∡퐵퐴퐶 = 50° Oscar: By definition of the theorem, we know that the opposite angle is equal: ∡퐴퐵퐶 = 50°, and the value of the angle ∡퐴퐶퐵 must be 80°, because that is the value we need to fulfill the sum of 180°. Miguel: Very good, now I can complete the homework assignment. Did you know that? There is a different way to classify triangles, which is as either acute or obtuse. The first one is characterized by having all their internal angles be less than 90°; and the last one by having one of the angles larger than 90°. ° Let´s get to work! Matching activity Up next, you will find the value of the non-congruent angle ∡퐵퐴퐶. Based on the fact that we are talking about an isosceles triangle, find the value of the congruent angles. Angle ∡푩푨푪 Value of congruent ∡ 50° 37.5° 10° 52.5° 25° 25° 130° 77.5° 75° 85° 105° 65° “Word blaster” activity Define the following concepts: Triangle Isosceles Sides Angles Congruent Fill in the blanks to complete the definition of an isosceles triangle A ______________ is ________________ if it has a pair of ______________ that are ______________, and two of its _________________ must also fulfill this condition. Activity 2 Right Triangles and their applications Miguel: During another class that you missed we covered the 30-60-90 and the Pythagorean Theorems. The first one is a particular case of right rectangles and the second one is a generalization. Oscar: What did you do during this class? Miguel: Teacher Alejandro began with characterization. First, you must distinguish the right triangles from the other. Figure. Triangles. Which ones are right triangles? It is possible to find them just by looking at them. Oscar: That is easy; they can be distinguished by the presence of a right angle, meaning 90°. For example, you can see that the purple and red triangles satisfy this requirement. Miguel: Very good, that is the idea. Now, what is the function of each theorem? The 30-60-90 theorem is applied to particular triangles, meaning those that possess this special characteristic, and it is used to find the value of each side, according to the following relationship: “The length of the hypotenuse is twice the length of the shortest cathetus, and the length of the longest cathetus is √3 times the length of the shortest cathetus” Look at an example: We need to find the height of an electric tower. We have the following information: the horizontal distance from the base of the tower to a reference point, as well as the angle formed from the reference point to the top of the tower, as shown in the figure. Oscar: I see. In this case we need to relate the distances, because if we extract the triangle from the figure, we get a 30-60-90 triangle. Then we develop the situation accordingly. Figure. Electric tower. Based on the relationship and the data acquired we have the following information: “The length of the longest leg is √3 times the length of the shortest leg”. Therefore we know that the height is the longest leg: 50 ∙ √3 = 86.6 ≈ 87푚. I know that you did not ask this, but the oblique distance, or hypotenuse, is 100m long, according to the definition of the theorem; “…twice the length of the shortest cathetus” Miguel: Very good! You have learned this quickly.