Chapter 8 Right Triangles and Trigonometry (2024)

Franklin County Community School Corporation ● FranklinCountyHigh School ● Brookville, Indiana

Curriculum Map

Course Title: Geometry Honors / Quarter: 4 / Academic Year: 2012-2013

Essential Questions for this Quarter:

  1. How do you solve problems using similarity relationships of right triangles?
  2. How do you apply trigonometric ratios to real-world problems?
  3. How do you develop and apply the properties of lines and angles that intersect circles?
  4. How do you apply the formulas for circles, polygons, and composite figures?
  5. How can you create representations of three-dimensional figures?
  6. How do you apply formulas for volume to real-world problems?

Unit/Time Frame / Standards / Content / Skills / Assessment / Resources
Chapter 8 Right Triangles and Trigonometry
8.1 Similarity in Right Triangles
8.2 Trigonometric Ratios
8.3 Trigonometric Ratios and Complementary Angles
8.4 Angles of Elevation and Depression
8.5Law of Sines and Law of Cosines
Chapter 12 Circles
12.1 Lines That Intersect Circles
12.2 Arcs and Chords
12.3 Sector Area and Arc Length
12.4 Inscribed Angles
12.5 Angle Relationships in Circles
12.6 Segment Relationships in Circles
Chapter 10 Extending Perimeter, Circumference, and Area
10.1 Developing Formulas for Triangles and Quadrilaterals
10.2 Developing Formulas for Triangles and Regular Polygons
10.3 Composite Figures
Chapter 11 Spatial Reasoning
11.1 Solid Geometry
11.2 Volume of Prisms and Cylinders
11.3 Volume of Pyramids and Cones
11.4Spheres / State Standards
G.2.5
G.2.17
G.2.19
G.2.21
G.3.2
G.3.3
G.3.4
G.3.5
G.4.1
G.4.3
G.4.4
Common Core Standards
CC.9-12. G.SRT.6
CC.9-12. G.SRT.7
CC.9-12. G.SRT.8
CC.9-12. G.SRT.10
CC.9-12. A.SSE.1
CC.9-12. G.GMD.1
CC.9-12. G.GMD.3
CC.9-12. G.GMD.4
CC.9-12. G.MG.3
CC.9-12. G.C.2
CC.9-12. G.C.5
CC.9-12. G.GPE.1
Standards for Mathematical Practice
SMP1
SMP2
SMP3
SMP4
SMP5
SMP6
SMP7
SMP8 / Similarity of right triangles.
How to use ratios and proportions to find missing side lengths in right triangles
How to use trigonometric ratios to solve real-world problems.
Solving problems involving circles
Finding lengths, angle measures, and areas associated with circles.
Applying circle theorems to solve a wide range of problems.
Areas and perimeters of figured whose vertices are given by ordered pairs.
Areas and perimeters of figures whose dimensions are found by using the Pythagorean Theorem.
Areas and perimeters of figures in customary and metric units.
Proofs of formulas for area and perimeter.
Properties of three-dimensional figures.
The volumes of three-dimensional figures. /

  • Use geometric mean to find segment lengths in right triangles.
  • Apply similarity relationships in right triangles to solve problems.
  • Find the sine, cosine, and tangent of an acute triangle.
  • Use trigonometric ratios to find side lengths in right triangles and to solve real-world problems.
  • Use trigonometric ratios to find the angle measures in right triangles and to solve real-world problems.
  • Solve problems involving angles of elevation and depression.
  • Use the Law of Sines and the Law of Cosines to solve triangles.
  • Identify Tangents, secants, and chords
  • Use properties of tangents to solve problems.
  • Apply properties of arcs.
  • Apply properties of chords.
  • Find the area of sectors.
  • Find arc lengths.
  • Find the measure of an inscribed angle.
  • Use inscribed angles and their properties to solve problems.
  • Find the measures of angles formed by lines that intersect circles.
  • Use angle measure to solve problems.
  • Find the lengths of segments formed by lines that intersect circles/.
  • Use the lengths of segments in circles to solve problems.
  • Develop and apply the formulas for the areas of triangles and special quadrilaterals.
  • Solve problems involving perimeters and areas of triangles and special quadrilaterals.
  • Develop and apply the formula for the area and circumference of a circle.
  • Develop and apply the formula for the area of a regular polygon.
  • Use the Arc Addition Postulate to find the area of composite figures.
  • Use composite figures to estimate the areas of irregular shapes.
  • Classify three-dimensional figures according to their properties.
  • Use nets and cross sections to analyze three-dimensional figures.
  • Learn and apply the formula for the volume of a prism.
  • Learn and apply the formula for the volume of a cylinder.
  • Learn and apply the formula for the volume of a pyramid.
  • Learn and apply the formula for the volume of a cone.
  • Learn and apply the formula for the volume of a sphere.
  • Learn and apply the formula for the surface area of a sphere.

/ Textbook: Holt McDougal Geometry Common Core 2012 Edition
Textbook Prentice Hall Geometry 1998 Edition
Textbook McDougal Littell Geometry 2004 Edition
Intro to Geometry Frank Schaffer Publications
Teacher generated worksheets
PowerPoint Presentations

Franklin County Community School Corporation ● FranklinCountyHigh School ● Brookville, Indiana

COMMON CORE AND INDIANA ACADEMIC STANDARDS

State Standards
G.1: Students find lengths and midpoints of line segments. They describe and use parallel and perpendicular lines. They find slopes and equations of lines.
G.1.1: Find the lengths and midpoints of line segments in one- or two-dimensional coordinate systems.
G.1.2: Construct congruent segments and angles, angle bisectors, and parallel and perpendicular lines using a straight edge and compass, explaining and justifying the process used.
G.1.3: Understand and use the relationships between special pairs of angles formed by parallel lines and transversals.
G.1.4: Use coordinate geometry to find slopes, parallel lines, perpendicular lines, and equations of lines.
G.2: Students identify and describe polygons and measure interior and exterior angles. They use congruence, similarity, symmetry, tessellations, and transformations. They find measures of sides, perimeters, and areas.
G.2.2: Find measures of interior and exterior angles of polygons, justifying the method used.
G.2.3: Use properties of congruent and similar polygons to solve problems.
G.2.4: Apply transformations (slides, flips, turns, expansions, and contractions) to polygons in order to determine congruence, similarity, symmetry, and tessellations. Know that images formed by slides, flips and turns are congruent to the original shape.
G.2.5: Find and use measures of sides, perimeters, and areas of polygons, and relate these measures to each other using formulas.
G.2.6: Use coordinate geometry to prove properties of polygons such as regularity, congruence, and similarity.
G.3: Students identify and describe simple quadrilaterals. They use congruence and similarity. They find measures of sides, perimeters, and areas.
G.3.1: Describe, classify, and understand relationships among the quadrilaterals square, rectangle, rhombus, parallelogram, trapezoid, and kite.
G.3.2: Use properties of congruent and similar quadrilaterals to solve problems involving lengths and areas.
G.3.3: Find and use measures of sides, perimeters, and areas of quadrilaterals, and relate these measures to each other using formulas.
G.3.4: Use coordinate geometry to prove properties of quadrilaterals such as regularity, congruence, and similarity.
G.4: Students identify and describe types of triangles. They identify and draw altitudes, medians, and angle bisectors. They use congruence and similarity. They find measures of sides, perimeters, and areas. They apply inequality theorems.
G.4.1: Identify and describe triangles that are right, acute, obtuse, scalene, isosceles, equilateral, and equiangular.
G.4.2: Define, identify, and construct altitudes, medians, angle bisectors, and perpendicular bisectors.
G.4.3: Construct triangles congruent to given triangles.
G.4.4: Use properties of congruent and similar triangles to solve problems involving lengths and areas.
G.4.5: Prove and apply theorems involving segments divided proportionally.
G.4.6: Prove that triangles are congruent or similar and use the concept of corresponding parts of congruent triangles.
G.4.7: Find and use measures of sides, perimeters, and areas of triangles, and relate these measures to each other using formulas.
G.4.8: Prove, understand, and apply the inequality theorems: triangle inequality, inequality in one triangle, and hinge theorem.
G.4.9: Use coordinate geometry to prove properties of triangles such as regularity, congruence, and similarity.
G.5: Students prove the Pythagorean Theorem and use it to solve problems. They define and apply the trigonometric relations sine, cosine, and tangent.
G.5.1: Prove and use the Pythagorean Theorem.
G.5.2: State and apply the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle.
G.5.4: Define and use the trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) in terms of angles of right triangles.
G.5.5: Know and use the relationship sin²x + cos²x = 1.
G.5.6: Solve word problems involving right triangles.
G.6: Students define ideas related to circles: e.g., radius, tangent. They find measures of angles, lengths, and areas. They prove theorems about circles. They find equations of circles.
G.6.2: Define and identify relationships among: radius, diameter, arc, measure of an arc, chord, secant, and tangent.
G.6.3: Prove theorems related to circles.
G.6.5: Define, find, and use measures of arcs and related angles (central, inscribed, and intersections of secants and tangents).
G.6.6: Define and identify congruent and concentric circles.
G.6.7: Define, find, and use measures of circumference, arc length, and areas of circles and sectors. Use these measures to solve problems.
G.6.8: Find the equation of a circle in the coordinate plane in terms of its center and radius.
G.7: Students describe and make polyhedra and other solids. They describe relationships and symmetries, and use congruence and similarity.
G.7.2: Describe the polyhedron that can be made from a given net (or pattern). Describe the net for a given polyhedron.
G.7.4: Describe symmetries of geometric solids.
G.7.5: Describe sets of points on spheres: chords, tangents, and great circles.
G.7.6: Identify and know properties of congruent and similar solids.
G.7.7: Find and use measures of sides, volumes of solids, and surface areas of solids, and relate these measures to each other using formulas.
G.8: Mathematical Reasoning and Problem Solving
G.8.6: Identify and give examples of undefined terms, axioms, and theorems, and inductive and deductive proof.
G.8.8: Write geometric proofs, including proofs by contradiction and proofs involving coordinate geometry. Use and compare a variety of ways to present deductive proofs, such as flow charts, paragraphs, and two-column and indirect proofs.
Common Core Standards
Congruence G-CO
Experiment with transformations in the plane
1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Understand congruence in terms of rigid motions
6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
Prove geometric theorems
9. Prove theorems about lines and angles. Theorems include: verticalangles are congruent; when a transversal crosses parallel lines, alternateinterior angles are congruent and corresponding angles are congruent;points on a perpendicular bisector of a line segment are exactly thoseequidistant from the segment’s endpoints.
10. Prove theorems about triangles. Theorems include: measures of interiorangles of a triangle sum to 180°; base angles of isosceles triangles arecongruent; the segment joining midpoints of two sides of a triangle isparallel to the third side and half the length; the medians of a trianglemeet at a point.
11. Prove theorems about parallelograms. Theorems include: oppositesides are congruent, opposite angles are congruent, the diagonalsof a parallelogram bisect each other, and conversely, rectangles areparallelograms with congruent diagonals.
Make geometric constructions
12. Make formal geometric constructions with a variety of tools andmethods (compass and straightedge, string, reflective devices,paper folding, dynamic geometric software, etc.). Copying a segment;copying an angle; bisecting a segment; bisecting an angle; constructingperpendicular lines, including the perpendicular bisector of a line segment;
and constructing a line parallel to a given line through a point not on theline.
13. Construct an equilateral triangle, a square, and a regular hexagoninscribed in a circle.
Similarity, Right Triangles, and Trigonometry G-SRT
Understand similarity in terms of similarity transformations
1. Verify experimentally the properties of dilations given by a center anda scale factor:
a. A dilation takes a line not passing through the center of thedilation to a parallel line, and leaves a line passing through thecenter unchanged.
b. The dilation of a line segment is longer or shorter in the ratiogiven by the scale factor.
2. Given two figures, use the definition of similarity in terms of similaritytransformations to decide if they are similar; explain using similaritytransformations the meaning of similarity for triangles as the equalityof all corresponding pairs of angles and the proportionality of allcorresponding pairs of sides.
3. Use the properties of similarity transformations to establish the AAcriterion for two triangles to be similar.
Prove theorems involving similarity
4. Prove theorems about triangles. Theorems include: a line parallel to oneside of a triangle divides the other two proportionally, and conversely; thePythagorean Theorem proved using triangle similarity.
5. Use congruence and similarity criteria for triangles to solve problemsand to prove relationships in geometric figures.
Define trigonometric ratios and solve problems involving righttriangles
6. Understand that by similarity, side ratios in right triangles areproperties of the angles in the triangle, leading to definitions oftrigonometric ratios for acute angles.
7. Explain and use the relationship between the sine and cosine ofcomplementary angles.
8. Use trigonometric ratios and the Pythagorean Theorem to solve righttriangles in applied problems.
Apply trigonometry to general triangles
9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle bydrawing an auxiliary line from a vertex perpendicular to the oppositeside.
10. (+) Prove the Laws of Sines and Cosines and use them to solveproblems.
11. (+) Understand and apply the Law of Sines and the Law of Cosinesto find unknown measurements in right and non-right triangles (e.g.,surveying problems, resultant forces).
Circles G-C
Understand and apply theorems about circles
1. Prove that all circles are similar.
2. Identify and describe relationships among inscribed angles, radii,and chords. Include the relationship between central, inscribed, andcirc*mscribed angles; inscribed angles on a diameter are right angles;the radius of a circle is perpendicular to the tangent where the radiusintersects the circle.
3. Construct the inscribed and circ*mscribed circles of a triangle, andprove properties of angles for a quadrilateral inscribed in a circle.
4. (+) Construct a tangent line from a point outside a given circle to thecircle.
Find arc lengths and areas of sectors of circles
5. Derive using similarity the fact that the length of the arc interceptedby an angle is proportional to the radius, and define the radianmeasure of the angle as the constant of proportionality; derive theformula for the area of a sector.
Expressing Geometric Properties with Equations G-GPE
Translate between the geometric description and the equation for a
conic section
1. Derive the equation of a circle of given center and radius using thePythagorean Theorem; complete the square to find the center andradius of a circle given by an equation.
2. Derive the equation of a parabola given a focus and directrix.
3. (+) Derive the equations of ellipses and hyperbolas given the foci,using the fact that the sum or difference of distances from the foci isconstant.
Use coordinates to prove simple geometric theorems algebraically
4. Use coordinates to prove simple geometric theorems algebraically. Forexample, prove or disprove that a figure defined by four given points in thecoordinate plane is a rectangle; prove or disprove that the point (1, √3) lieson the circle centered at the origin and containing the point (0, 2).
5. Prove the slope criteria for parallel and perpendicular lines and usethem to solve geometric problems (e.g., find the equation of a lineparallel or perpendicular to a given line that passes through a givenpoint).
6. Find the point on a directed line segment between two given pointsthat partitions the segment in a given ratio.
7. Use coordinates to compute perimeters of polygons and areas oftriangles and rectangles, e.g., using the distance formula.
Geometric Measurement and Dimension G-GMD
Explain volume formulas and use them to solve problems
1. Give an informal argument for the formulas for the circumference ofa circle, area of a circle, volume of a cylinder, pyramid, and cone. Usedissection arguments, Cavalieri’s principle, and informal limit arguments.
2. (+) Give an informal argument using Cavalieri’s principle for theformulas for the volume of a sphere and other solid figures.
3. Use volume formulas for cylinders, pyramids, cones, and spheres tosolve problems.
Visualize relationships between two-dimensional and threedimensional
objects
4. Identify the shapes of two-dimensional cross-sections of threedimensionalobjects, and identify three-dimensional objects generatedby rotations of two-dimensional objects.
Modeling with Geometry G-MG
Apply geometric concepts in modeling situations
1. Use geometric shapes, their measures, and their properties to describeobjects (e.g., modeling a tree trunk or a human torso as a cylinder).
2. Apply concepts of density based on area and volume in modelingsituations (e.g., persons per square mile, BTUs per cubic foot).
3. Apply geometric methods to solve design problems (e.g., designingan object or structure to satisfy physical constraints or minimize cost;working with typographic grid systems based on ratios).
Standards for Mathematical Practice
SMP1. Make sense of problems and persevere in solving them.
SMP2. Reason abstractly and quantitatively.
SMP3. Construct viable arguments and critique the reasoning of others.
SMP4. Model with mathematics.
SMP5. Use appropriate tools strategically.
SMP6. Attend to precision

Chapter 8 Right Triangles and Trigonometry (2024)

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