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1 BELL WORK

2 Geometry Day 52 Topic: Assessment 2.1 Chapter

3 Chapter 8 Big Ideas Measurement Some attributes of geometric figures, such as length, area, volume, and angle measure, are measurable. Units are used to describe these attributes. Similarity Two geometric figures are similar when corresponding lengths are proportional and corresponding angles are congruent. In this chapter, you will: Explore the Pythagorean Theorem and its Converse Explore Special Triangles and the relationships of their sides/angles Explore the Trigonometric Ratios (of right triangles)

4 Common Core State Standards Geometry (GM) Similarity, Right Triangles, and Trigonometry GM: G.SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

5 Chapter 8 Essential Understanding 8-1 If the lengths of any two sides of a right triangle are known, the length of the third side can be found by using the Pythagorean Theorem. 8-2 Certain Right Triangles have properties that allow their side lengths to be determined without using the Pythagorean Theorem. 8-3 If certain combinations of side lengths and angle measures of a right triangle are known, ratios can be used to find other side lengths and angle measures. 8-4 The angles of elevation and depression are the acute angles of right triangles formed by a horizontal distance and a vertical height.

6 Student Objectives I can use the Pythagorean Theorem and its converse. (8-1) I can use the properties of 45⁰ 45⁰ 90⁰ and 30⁰ 60⁰ 90⁰ triangles. I can use the Sine, Cosine, and Tangent ratios to determine side lengths and angle measures in Right Triangles. I can use Angles of Elevation and Depression to solve problems.

7 The Pythagorean Theorem & Converse Theorem 8-1 Pythagorean Theorem If a triangle is a right triangle, then the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. Pythagorean Triple A Pythagorean Triple is a set of nonzero whole numbers (natural numbers) a, b, and c that satisfy the equation a 2 + b 2 = c 2 Theorem 8-2 Converse of the Pythagorean Theorem If the sum of the squares of the lengths of the two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle.

8 Additional Theorems (Obtuse/Acute) Theorem 8-3 If the square of the length of the longest side of a triangle is greater than the sum of the squares of the length of the other two sides, then the triangle is Obtuse. Theorem 8-4 If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is acute.

9 Theorem 8-5 p.499 Theorem Triangle Theorem In a triangle, both legs are congruent and the length of the hypotenuse is 2 times the length of a leg (AKA: Isosceles Right Triangle)

10 Theorem 8-6 p.501 Theorem Triangle Theorem In a triangle, the length of the hypotenuse is twice the length of the shorter leg. The length of the longer leg is 3 times the length of the shorter leg.

11 Trigonometry S O H CA H TO A Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies.

12 Chapter 8-3 Vocabulary S O H CA H TO A Trigonometric Ratios For any right triangle, there are six trig ratios: Sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). Sine Sin = Cosine Cos = Opposite Hypotenuse Adjacent Hypotenuse Tangent Tan = Opposite Adjacent

13 All Six Ratios S O H CA H TO A

14 Using Trigonometric Ratios S O H CA H TO A When solving for a side length Decide on the ratio to use based on side length and angle measure you are given and the side length you are seeking Set up an equation using the Trig ratio with respect to that measure and the ratio that coincides using a variable of your choice for the unknown side length tan 42 = x 12 X 42 12

15 Using Trigonometric Ratios S O H CA H TO A When solving for an angle measure Decide on the ratio to use based on side lengths you are given and the angle measure you are seeking Set up an equation using the Trig ratio with respect to that angle measure and the ratio that coincides using a variable of your choice for the unknown angle measure cos X = 8 10 ***Use the inverse Trig button when solving for X*** 10 X = cos X 8

16 Angle of Elevation / Depression An angle of elevation (depression) is the angle formed by a horizontal line and the line of sight to an object above (below) the horizontal line. (Glossary term p. 912) The angle of elevation of an object as seen by an observer is the angle between the horizontal and the line from the object to the observer's eye (the line of sight).

17 Angle of Elevation / Depression

18 Law of Sines The Law of Sines is the relationship between the sides and angles of non-right (oblique) triangles. Simply, it states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all sides and angles in a given triangle.

19 Law of Cosines The law of cosines for calculating one side of a triangle when the angle opposite and the other two sides are known. Can be used in conjunction with the law of sines to find all sides and angles. Click on the highlighted text for either side c or angle C to initiate calculation.

20 Testing Procedures Take out at least two #2 lead pencils. You may only keep your Learning Log (Composition Notebook) out on your desk. You will maintain silence throughout the entire test. If you have a question, raise your hand and I will come to you. Once you complete the test, you will turn it in to me and immediately return to your seats. Upon completion, you will have instructions on the board.

21 Test in Progress!!! Remain seated Remain quiet Eyes on your own paper Good Luck!!!

22 Test Timer You will be given 50 minutes to complete this assessment. At the end of that time, we will begin with a brief introduction to Chapter 10

23 Chapter 10 Big Ideas Measurement Some attributes of geometric figures, such as length, area, volume, and angle measure, are measurable. Units are used to describe these attributes. Similarity Two geometric figures are similar when corresponding lengths are proportional and corresponding angles are congruent. In this chapter, you will: Explore the Area of Parallelograms and Triangles Explore the Area of Trapezoids, Rhombuses, and Kites Explore the Area of Regular Polygons Explore Perimeters and areas of Similar Figures Explore Trigonometry and Area Explore Circle and Arcs

24 Chapter 10 Essential Understanding 10-1 The area of a parallelogram or a triangle can be found when the length of its base and its height are known The area of a trapezoid can be found when the height and the lengths of its base are known. The area of a rhombus or kite can be found when the length of its diagonals are known The area of a regular polygon is a function of the distance from center to a side and the perimeter Ratios can be used to compute the perimeters and areas of similar figures Trigonometry can be used to find the area of a regular polygon when the length of a side, radius, or apothem is known or to find the area of a triangle when the length of two sides and the included angle is known The length of a part of a circle s circumference can be found by relating it to an angle in the circle The area of parts of a circle formed by radii and arcs can be found when the circle s radius is known.

25 Common Core State Standards Geometry (GM) Apply geometric concepts in modeling situations. GM:MG.A.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Use coordinates to prove simple geometric theorems algebraically. GM:GPE.B.7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

26 Student Objectives I can find the measures of central angles and arcs. I can find the circumference and arc length.

27 Khan Academy

28 Chapter 10 Area Open your textbooks to Chapter 10.6 (p.649) You will begin by writing the new vocabulary terms found in this section: Circle Center Diameter Radius Congruent Circles Central Angle Semicircle Minor arc Major arc Adjacent arcs Circumference Pi Concentric circles Arc length

29 To be continued S O H CA H TO A Tomorrow, Thursday 3/30/2017, we will begin Chapter 10 and quickly move on to Chapter 12. Your next test 2.2 (10-6, 12-1, 2, 3, & 4) will be on Thursday 06 April 2017 Homework: N/A Visit our website for resources to help prepare you for your test.

30 Any Questions???

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