Chapter 6 Proportions and Similarity Ananth Dandibhotla, William Chen, Alden Ford, William Gulian Key Vocabulary Proportion An equality statement with 2
ratios Cross Products a*d and b*c, in a/b = c/d Similar Polygons Polygons with the same shape Scale Factor A ratio comparing the sizes of similar polygons Midsegment A line segment connecting the midpoints of two sides of a triangle 6-1 Proportions Ratios compare two values, a/b, a:b (b 0)
For any numbers a and c and any non-zero number numbers b and d: a/b = c/d iff ad = bc Ratio s Problem Bob made a 18 in. x 20 in. model of a famous painting. If the original paintings dimensions are 3ft x a ft, find a.
Answer: a = 10/4 4 6-2 Similar Polygons Polygons with the same shape are similar polygons ~ means similar Scale factors compare the lengths of corresponding pieces of a polygon
Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding angles are 2 : 1 proportional. The order of the points matters Problem
ABC and DEF have the same angle measures. Side AB is 2 units long Side BC is 10 units long Side DE is 3 units long Side FD is 15 units long Are the triangles similar? Answer: They are not similar. 6
6-3 Similar Triangles Identifying Similar Triangles: AA~ -Postulate- If the two angles of one triangle are congruent to two angles of another triangle, then the triangles are ~ SSS~ -Theorem- If the measures of the corresponding sides of two triangles are proportional, then the triangles are ~ SAS~ -Theorem- If the measures of two sides of a
triangle are proportional to the measures of two corresponding sides of another triangle and the included angles are congruent, the triangles are ~ 6-3 Similar Triangles (cont.) Theorem 6.3 similar triangles are reflexive, symmetric, and transitive SSS
AA SAS Problem Determine whether each pair of triangles is similar and if so how? Answer: They are similar by the SSS Similarity
9 6-4 Parallel Lines and Proportional Parts Triangle Proportionality Theorem If a line is parallel to one side of a triangle and intersects the other two sides in two distinct point, then it separates these sides into segments of proportional length Tri. Proportion Thm. Converse If a line
intersects two sides of a triangle and separates the sides into corresponding segments of proportional lengths, then the line is parallel to the third side 6-4 Parallel Lines and Proportional Parts (Cont.) Midsegment is a segment whose endpoints are the midpoints of 2 sides of a triangle. Midsegment Thm: A midsegment of a triagnle
is parallel to one side of the triangle , and its length is one- half the length of that side. Corollary 6.1: If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally. Corollary 6.2: If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. 11
Problem Find x and ED if AE = 3, AB = 2, BC = 6, and ED = 2x - 3 Answer: x = 6 and ED =9 12 6-5 Parts of Similar Triangles
Proportional Perimeters Thm. If two triangles are similar, then the perimeters are proportional to the measures of corresponding sides Thm 6.8-6.10 triangles have corresponding (altitudes/angle bisectors/medians) proportional to the corresponding sides Angle Bisector Thm. An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other
two sides Problem Find the perimeter of DEF if ABC ~ DEF, Ab = 5, BC = 6, AC = 7, and DE = 3. Answer: The perimeter is 10.8 14
Wacaw Sierpiski and his Triangle 1882-1969, Warsaw, Poland A mathematician, Sierpiski studied in the Department of Mathematics and Physics, at the University of Warsaw in 1899. Graduating in 1904, he became a teacher of the subjects. The Triangle: If you connect the midpoints of the sides of an equilateral triangle, itll form a smaller triangle. In the three triangular spaces, you can create more triangles by repeating the process, indefinitely. This example of a fractal (geometric figure created by iteration, or repeating the same procedure over and over again) was described by Sierpiski, in 1915. Other Sierpiski fractals: Sierpiski Carpet, Sierpiski Curve Other contributions: Sierpiski numbers, Axiom of Choice, Continuum hypothesis
Completely unrelated: Theres a crater on the moon named after him. 15 Time Left? 6-6 Fractals!
RIOO-RS system will be available mid-December - March for training and testing. Verify data. Updates available for most frequently changed items first. Data will be updated and over written with production data on a regular interval. RIOO-RS system will continue...
Ms. Kathy Edwards is a Certified Occupational Health Nurse Specialist. She served as the Department Head for Occupational Medicine at Naval Medical Center San Diego for 20 years (1987-2007) and was a Navy nurse for 10 years prior to that.
Systolic Click. Very sharp, high frequency click of Mitral valve prolapse, in early MVD. Snapping of the chordae tendinae as they go taught. PMI left apex. Mid-Systolic (audio) (audio) May be accompanied by a systolic murmur. Early, late, or holosystolic....
Lots of Farm Real Estate Transactions ... Ottawa, IL 61350 (815) 433-1780. Region 3. Herbert Meyer, ARA. 1. st. Farm Credit Services. ... Estate Taxes - Rising farmland values are putting significant pressure on the estate tax exemption and are...
Caution! Irrational numbers can be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers. If a whole number is not a perfect square, then its square root is an...
pNeuton-Notfallrespirator mini Respirator Eigenständige Tour Klinische Fragen MRT-Einsatz pNeuton ist kompatibel für den Einsatz mit MRT-Scannern mit einer Feldstärke bis zu 3 Tesla Tests belegen zwar, dass der pNeuton im Inneren des MRT-Scanners funktioniert, jedoch wird empfohlen, einen Mindestabstand von...
CHAPTER 14: THEORIES OF PERSONALITY. The thing that makes us think, feel, and act differently. PERSONALITY. ... viewing oneself in a positive light due to positive feedback from others (Unconditional positive regard) CONDITIONS OF WORTH: conditions a person must meet...
Mitigating Potential Bias. The content of this program was developed by the planning committee and peer reviewed by Thrombosis Canada. Recommendations involving clinical medicine are based on evidence that is accepted within the profession; and all scientific research referred to,...
Ready to download the document? Go ahead and hit continue!